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?
To determine the area of the third square, we need to find the length of the corresponding side of the triangle.
Let's denote the length of the sides of the triangle as a, b, and c, with a being the length of the side corresponding to Square A, b being the length corresponding to the unknown square, and c being the length corresponding to Square C.
According to the problem, the area of Square A is 25 units^2. Therefore, the side length a can be found by taking the square root of 25:
a = √25 = 5 units
Similarly, the area of Square C is 70 units^2. Therefore, the side length c can be found by taking the square root of 70:
c = √70
Now, in a right-angled triangle, the squares of the lengths of the two shorter sides added together must equal the square of the length of the longest side. This is known as the Pythagorean theorem:
a^2 + b^2 = c^2
Substituting the known values, we have:
(5^2) + b^2 = (√70)^2
25 + b^2 = 70
b^2 = 70 - 25 = 45
b = √45 = √(9 x 5) = 3√5 units
Therefore, the length of the third side of the triangle (corresponding to the unknown square) is 3√5 units. To find the area of the third square, we square this length:
Area of the third square = (3√5)^2 = 3^2 × (√5)^2 = 9 × 5 = 45 units^2
So, the area of the third square must be 45 units^2 for the triangle to have a right angle.
To determine the area of the third square, we need to find the length of the corresponding side of the triangle.
Let us label the lengths of the sides of the triangle, with side A corresponding to square A and side C corresponding to square C.
Given that the area of square A is 25 units², we can find the length of side A by taking the square root of the area. So, the length of side A is √25 = 5 units.
Similarly, given that the area of square C is 70 units², we can find the length of side C by taking the square root of the area. So, the length of side C is √70 units.
To determine whether the triangle has a right angle, we can use the Pythagorean theorem, which states that in a right 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 lengths of the other two sides.
Applying this theorem, we have:
(side A)² + (side B)² = (side C)²
Since we know the lengths of side A and side C, we can substitute those values into the equation:
(5 units)² + (side B)² = (√70 units)²
25 units² + (side B)² = 70 units²
Now, we can solve for the length of side B:
(side B)² = 70 units² - 25 units²
(side B)² = 45 units²
Taking the square root of both sides, we find:
side B = √45 units
So, if the triangle has a right angle, the length of side B would be √45 units.
Now, to find the area of the third square, we need to square the length of side B:
Area of square B = (side B)²
Area of square B = (√45 units)²
Area of square B = 45 units²
Therefore, the area of the third square would be 45 units² for the triangle to have a right angle.
To find the area of the third square, we first need to determine the sides of the triangle. Let's assume that the sides of the triangle are a, b, and c.
We know that Square A has an area of 25 units², which means its side length is √25 = 5 units. Similarly, Square C has an area of 70 units², so its side length is √70 units.
To determine if the triangle has a right angle, we can use the Pythagorean theorem. According to the theorem, 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.
Based on the above information, we can set up the equation:
a² + b² = c²
Plugging in the side lengths from Square A and Square C, we get:
5² + b² = (√70)²
25 + b² = 70
b² = 70 - 25
b² = 45
Taking the square root of both sides, we find:
b = √45
Now, to find the side length of the third square, we need to calculate the square root of 45. Once we have the side length, we can square it to find the area of the third square.
Hence, the area of the third square for the triangle to have a right angle can be found by calculating (√45)².