A triangle is drawn and squares are created from the lengths of each side.
Square A has an area of 25 units² and Square C has an area of 70 units².
What must be the area of the 3rd square for triangle to have a right angle? Click Here for Help Video.
(1 point)
The area of square B is
units².
To determine the area of square B, we must first find the lengths of the sides of the triangle.
Let's assume that the sides of the triangle are x, y, and z.
The area of square A is given as 25 units², hence its side length is √25 = 5 units.
This means that the side x of the triangle is 5 units.
Similarly, the area of square C is given as 70 units², hence its side length is √70 units.
This means that the side z of the triangle is √70 units.
Now, to find the side length y of the triangle, we can use the Pythagorean theorem:
x² + y² = z²
5² + y² = (√70)²
25 + y² = 70
y² = 45
y = √45
Now that we have the lengths of all three sides of the triangle, we can find the area of square B, which is created from the length of side y.
The area of square B is y² = (√45)² = 45 units².
Therefore, the area of square B is 45 units².
To find the area of Square B, we need to determine the length of one side of the triangle.
Since Square A has an area of 25 units², its side length is the square root of 25, which is 5 units.
Similarly, since Square C has an area of 70 units², its side length is the square root of 70.
To determine if the triangle has a right angle, we need to see if the sum of the squares of the two smaller side lengths is equal to the square of the longest side length.
Let's assume that Square B has an area of x units². The side length of Square B would then be the square root of x.
To check if the triangle has a right angle, we can use the Pythagorean theorem.
The Pythagorean theorem states that for a right-angled triangle, the square 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 hypotenuse is the side of the triangle corresponding to Square C, which has a length of the square root of 70 units.
So, using the Pythagorean theorem, we can write the equation:
(Square root of 70)² = (5)² + (Square root of x)²
70 = 25 + x
Simplifying the equation:
70 - 25 = x
45 = x
Therefore, the area of Square B is 45 units².
To find the area of square B, we need to determine the length of one of the sides of the triangle. Since squares A and C have known areas, we can find their side lengths by taking the square root of their areas.
Given that square A has an area of 25 units², we can take the square root of 25 to find the length of its sides. The square root of 25 is 5, so square A has a side length of 5 units.
Similarly, square C has an area of 70 units². Taking the square root of 70 gives us approximately 8.37. However, we need to consider that the side length of a square cannot be a decimal value. Therefore, we can round 8.37 to the nearest whole number, which is 8. Hence, square C has a side length of 8 units.
Next, let's examine the relationship between the side lengths of the triangle and the area of square B. In a right-angled triangle, the squares of the two shorter sides (legs) will be equal to the square of the longest side (hypotenuse). This is known as the Pythagorean theorem.
Since we know the side lengths of squares A and C, we can calculate the lengths of the legs of the triangle. Square A has a side length of 5 units, which corresponds to the shorter side of the triangle. Square C has a side length of 8 units, which corresponds to the longer side of the triangle.
Using the Pythagorean theorem, we can solve for the length of the remaining side. Let's denote the side length of square B as "x". According to the Pythagorean theorem, we have the equation:
5² + x² = 8²
25 + x² = 64
x² = 39
Taking the square root of both sides, we find:
x ≈ 6.24
Again, since a square cannot have a side length with decimal values, we round 6.24 to the nearest whole number. Therefore, the side length of square B is 6 units.
Finally, to find the area of square B, we square its side length:
Area of square B = 6² = 36 units².
Thus, the area of square B is 36 units².