Question

1. use the pythagorean theorem to solve for the unknown side lengths

a) solve for x
b) solve for y

Explanation:

Part (a): Solve for \( x \)

Step 1: Recall the Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), \( a^2 + b^2 = c^2 \). Here, the legs are \( 6 \) and \( 8 \), and the hypotenuse is \( x \). So we substitute \( a = 6 \), \( b = 8 \), and \( c = x \) into the formula:
\( 6^2 + 8^2 = x^2 \)

Step 2: Calculate the squares

Calculate \( 6^2 = 36 \) and \( 8^2 = 64 \). Then add them together:
\( 36 + 64 = x^2 \)
\( 100 = x^2 \)

Step 3: Solve for \( x \)

Take the square root of both sides. Since \( x \) represents a length, we take the positive square root:
\( x = \sqrt{100} \)
\( x = 10 \)

Part (b): Solve for \( y \)

Step 1: Recall the Pythagorean Theorem

For a right triangle, \( a^2 + b^2 = c^2 \). Here, the hypotenuse \( c = 13 \), one leg \( a = 12 \), and the other leg is \( y \). So we substitute \( a = 12 \), \( c = 13 \), and \( b = y \) into the formula:
\( 12^2 + y^2 = 13^2 \)

Step 2: Calculate the squares

Calculate \( 12^2 = 144 \) and \( 13^2 = 169 \). Then rearrange the formula to solve for \( y^2 \):
\( y^2 = 13^2 - 12^2 \)
\( y^2 = 169 - 144 \)

Step 3: Solve for \( y \)

Subtract the values:
\( y^2 = 25 \)
Take the square root of both sides. Since \( y \) represents a length, we take the positive square root:
\( y = \sqrt{25} \)
\( y = 5 \)

Answer:

s:
a) \( x = 10 \)
b) \( y = 5 \)