Pythagorean Theorem Proofs Practice
Complete this assessment to review what you've learned. It will not count toward your grade.
1 of 51 of 5 Items
Question
Use the image to answer the question.
A right-angled triangle has the vertices labeled upper P upper S upper T. The height upper S upper T is labeled p; the base upper T upper P is labeled s; and the hypotenuse upper P upper S is labeled t.
Which equation is a true equation based on the right triangle shown?
Option 1: p2+t2=s2
Option 2: p2+s2=t2
Option 3: There is not enough information about the triangle to write an equation.
Enter "1", "2", or "3".
(1 point)
wrong
Option 2: p2+s2=t2
The correct answer is Option 2: p^2 + s^2 = t^2.
To answer this question, we need to use the Pythagorean Theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In the given triangle, let's label the hypotenuse as 'c' instead of 't', to follow the traditional notation. So we have:
c^2 = a^2 + b^2
In the given options, we have p as the height (opposite the right angle), s as the base, and t as the hypotenuse. So we need to figure out which equation represents the Pythagorean Theorem in terms of p, s, and t.
Let's compare the options to the Pythagorean Theorem equation:
Option 1: p^2 + t^2 = s^2
Option 2: p^2 + s^2 = t^2
Comparing these options to the Pythagorean Theorem equation, we can see that Option 2 matches:
p^2 + s^2 = t^2
Therefore, the correct answer is Option 2.
If a triangle has sides 12, 16, and 20, which length is the length of the hypotenuse?(1 point)
To determine the length of the hypotenuse, we can use the Pythagorean Theorem.
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the sides are 12, 16, and 20. We can label the hypotenuse as 'c', and the other two sides as 'a' and 'b'. We want to find the length of the hypotenuse, which is 'c'.
Using the Pythagorean Theorem, we have:
c^2 = a^2 + b^2
Plugging in the values, we have:
c^2 = 12^2 + 16^2
c^2 = 144 + 256
c^2 = 400
Taking the square root of both sides, we have:
c = √400
c = 20
Therefore, the length of the hypotenuse is 20 units.
Triangle GHI
is a right triangle. The length of one leg is 11 and the hypotenuse is 22. What is the length of the other leg? Round your answer to two decimal places.(1 point)
I apologize for the mistake. Let's solve the problem correctly.
According to the Pythagorean Theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
Let's label the legs of the right triangle as 'a' and 'b', and the hypotenuse as 'c'.
Given:
Length of leg a: 11
Length of hypotenuse c: 22
Using the Pythagorean Theorem, we can set up the following equation:
a^2 + b^2 = c^2
Plugging in the known values, we have:
11^2 + b^2 = 22^2
Simplifying further:
121 + b^2 = 484
Subtracting 121 from both sides:
b^2 = 363
Taking the square root of both sides:
b ≈ ±19.07
Since the length of a side cannot be negative, the length of the other leg is approximately 19.07 units (rounded to two decimal places).
Therefore, the length of the other leg is approximately 19.07 units.
Triangle GHI is a right triangle. The length of one leg is 11 and the hypotenuse is 22. What is the length of the other leg? Round your answer to two decimal places.(1 point)
Which statement explains the Pythagorean Theorem?(1 point)
Responses
If a triangle is a right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse.
If a triangle is a right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse.
If the interior angles of a triangle add up to 180°, then the sum of the legs of the triangle is equal to the square of the hypotenuse.
If the interior angles of a triangle add up to 180 degrees , then the sum of the legs of the triangle is equal to the square of the hypotenuse.
If a triangle is a right triangle, then the sum of the squares of the longest leg and the hypotenuse is equal to the square of the third side.
If a triangle is a right triangle, then the sum of the squares of the longest leg and the hypotenuse is equal to the square of the third side.
If all angles in a triangle are of equal measure, then the length of all sides will be equal.
The correct statement that explains the Pythagorean Theorem is:
If a triangle is a right triangle, then the sum of the squares of the legs is equal to the square of the hypotenuse.
Pythagorean Theorem Proofs Quick Check
2 of 52 of 5 Items
Question
Use the image to answer the question.
A triangle has the vertices labeled upper E upper F upper D. The height upper E upper F is labeled 15, the base upper D upper F is labeled 20, and the hypotenuse upper D upper E is labeled 25. Angle upper E measures 60 degrees and angle upper D measures 30 degrees.
Does the Pythagorean Theorem apply to this triangle? How do you know?
(1 point)
Responses
No, because the angle at point F is 90°.
No, because the angle at point cap f is 90 degrees .
Yes, because it is a right triangle.
Yes, because it is a right triangle.
Yes, because you can use the Pythagorean Theorem on any triangle
Yes, because you can use the Pythagorean Theorem on any triangle
No, because the triangle is not a right triangle.
The correct response is:
No, because the triangle is not a right triangle.
In the given triangle, it is stated that the angle at point F is 90 degrees. The Pythagorean Theorem applies to right triangles specifically, where one of the angles is a right angle (90 degrees). Since the given triangle does not have a right angle, the Pythagorean Theorem does not apply to it.
Using the Converse of the Pythagorean Theorem, explain whether a triangle with sides 15, 10, and 12 is a right triangle.(1 point)
Responses
No, because 22 is not equal to 15.
No, because 22 is not equal to 15.
No, because 244 is not equal to 225.
No, because 244 is not equal to 225.
No, because 325 is not equal to 144.
No, because 325 is not equal to 144.
No, because 369 is not equal to 100.
To determine whether a triangle is a right triangle, we can use the Converse of the Pythagorean Theorem.
The Converse of the Pythagorean Theorem states that if the sum of the squares of the lengths of the two shorter sides of a triangle is equal to the square of the length of the longest side, then the triangle is a right triangle.
In this case, we have the lengths of the sides of the triangle as 15, 10, and 12.
Calculating the squares of the lengths of the sides, we have:
15^2 = 225
10^2 = 100
12^2 = 144
Now we need to check if the sum of the squares of the two shorter sides is equal to the square of the longest side:
100 + 144 = 244
Since 244 is not equal to 225, the triangle with sides 15, 10, and 12 is not a right triangle.
Therefore, the correct response is:
No, because 244 is not equal to 225.
If the equation 182 + b2 = 302 is found to be true, what do we know about the triangle?(1 point)
Responses
The triangle is a right triangle, and the legs are 30 and 24.
The triangle is a right triangle, and the legs are 30 and 24.
The triangle is a right triangle, and the hypotenuse is 18.
The triangle is a right triangle, and the hypotenuse is 18.
The triangle is a right triangle with a missing leg length of 24.
The triangle is a right triangle with a missing leg length of 24.
The triangle is a right triangle with a missing side of 34.99.
If the equation 182 + b^2 = 302 is found to be true, we know that the triangle is a right triangle with a missing leg length of 24.
This is because the equation matches the Pythagorean Theorem, which states that in a right-angled triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
In this case, the equation 182 + b^2 = 302 can be rearranged as b^2 = 120, which means that the square of the missing leg length is 120. Taking the square root of both sides, we find that the missing leg length is approximately 10.95 (rounded to two decimal places), which is equal to 24 when squared.
Therefore, we know that the triangle is a right triangle with a missing leg length of 24.
The correct response is:
The triangle is a right triangle with a missing leg length of 24.
What is a converse of a theorem that is an if-then statement?(1 point)
Responses
The converse of a theorem has no relationship to the original theorem.
The converse of a theorem has no relationship to the original theorem.
The two parts are negated by using the word not.
The two parts are negated by using the word not .
The if part and the then part switch places.
The if part and the then part switch places.
The converse is the same as the original theorem.
The converse of a theorem that is an if-then statement is when the if part and the then part switch places.
In an if-then statement, the "if" part is the hypothesis or condition, and the "then" part is the conclusion. The converse of that statement swaps the positions of the hypothesis and the conclusion.
For example, if the original theorem states "If a triangle has equal sides, then it is equilateral", the converse of that theorem would be "If a triangle is equilateral, then it has equal sides."
Therefore, the correct response is:
The if part and the then part switch places.
To find the length of the other leg in a right triangle, we can use the Pythagorean Theorem.
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, let's label the legs of the right triangle as 'a' and 'b', and the length of the hypotenuse as 'c'.
So we have the following information:
Length of leg a: 11
Length of hypotenuse c: 22
Using the Pythagorean Theorem, we can set up the following equation:
a^2 + b^2 = c^2
Plugging in the known values, we have:
11^2 + b^2 = 22^2
Simplifying further:
121 + b^2 = 484
Subtracting 121 from both sides:
b^2 = 363
Taking the square root of both sides:
b = √363
b ≈ 19.06 (rounded to two decimal places)
Therefore, the length of the other leg is approximately 19.06 units.
To find the length of the other leg of a right triangle, we can use the Pythagorean Theorem.
The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, let's label the legs of the right triangle as 'a' and 'b', and the length of the hypotenuse as 'c'.
Given:
Length of leg a: 11
Length of hypotenuse c: 22
Using the Pythagorean Theorem, we can set up the following equation:
a^2 + b^2 = c^2
Plugging in the known values, we have:
11^2 + b^2 = 22^2
Simplifying further:
121 + b^2 = 484
Subtracting 121 from both sides:
b^2 = 363
Taking the square root of both sides:
b ≈ ±19.07
Since the length of a side cannot be negative, the length of the other leg is approximately 19.07 units (rounded to two decimal places).
Therefore, the length of the other leg is approximately 19.07 units.