Question
7
A triangle is drawn and squares are created from the lengths of each side.
Square A has an area of
17
units
2
17 units
2
and Square B has an area of
35
units
2
.
35 units
2
.
What must be the area of the 3rd square for triangle to have a right angle?
The area of Square C is
units
2
.
units
2
.
A triangle is drawn and squares are created from the lengths of each side.
Square A has an area of
17
units
2
17 units
2
and Square B has an area of
35
units
2
.
35 units
2
.
What must be the area of the 3rd square for triangle to have a right angle?
The area of Square C is
units
2
.
units
2
.
Answers
GPT-4o mini
In a right triangle, the relationship between the areas of the squares constructed on the sides is given by the Pythagorean theorem. This theorem states that for a right triangle with sides \(a\), \(b\), and \(c\) (where \(c\) is the hypotenuse), the following equation holds:
\[
a^2 + b^2 = c^2
\]
In your case, we have:
- The area of Square A (side \( a \)) is \( 17 \) units\(^2\) which means \( a^2 = 17 \).
- The area of Square B (side \( b \)) is \( 35 \) units\(^2\) which means \( b^2 = 35 \).
- We need to find the area of Square C (side \( c \)), which corresponds to side \( c \) of the triangle.
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Substituting the values:
\[
17 + 35 = c^2
\]
\[
52 = c^2
\]
Thus, the area of Square C is \( 52 \) units\(^2\).
So, the area of Square C is \( 52 \) units\(^2\).
\[
a^2 + b^2 = c^2
\]
In your case, we have:
- The area of Square A (side \( a \)) is \( 17 \) units\(^2\) which means \( a^2 = 17 \).
- The area of Square B (side \( b \)) is \( 35 \) units\(^2\) which means \( b^2 = 35 \).
- We need to find the area of Square C (side \( c \)), which corresponds to side \( c \) of the triangle.
Using the Pythagorean theorem:
\[
a^2 + b^2 = c^2
\]
Substituting the values:
\[
17 + 35 = c^2
\]
\[
52 = c^2
\]
Thus, the area of Square C is \( 52 \) units\(^2\).
So, the area of Square C is \( 52 \) units\(^2\).