Question

start by substituting the values you know and the value you dont know into the equation. the pythagorean theorem says that if a, b, and c are the sides of a right triangle, where c is the hypotenuse, then: $a^2 + b^2 = c^2$. in the right triangle, $a = 6$, $c = 10$, $b =?$; and there is an equation fill - in: $\square^2+\square^2 = \square^2$ with options 6, b, c, 10.

Explanation:

Step1: Identify known values

From the right triangle, \( a = 6 \), \( c = 10 \), and we need to find \( b \). The Pythagorean theorem is \( a^2 + b^2 = c^2 \). Substitute \( a = 6 \), \( c = 10 \) into the formula.
So the first box (for \( a \)) is \( 6 \), the third box (for \( c \)) is \( 10 \), and the second box is \( b \). Thus, the equation becomes \( 6^2 + b^2 = 10^2 \).

Step2: Solve for \( b^2 \)

Calculate \( 6^2 = 36 \) and \( 10^2 = 100 \). Then \( 36 + b^2 = 100 \). Subtract \( 36 \) from both sides: \( b^2 = 100 - 36 = 64 \).

Step3: Solve for \( b \)

Take the square root of both sides: \( b = \sqrt{64} = 8 \). But for the substitution step (the boxes), we just need to fill \( 6 \), \( b \), \( 10 \) in the respective boxes. Wait, the question here is about substituting into the equation first. So the substitution is \( 6^2 + b^2 = 10^2 \). So the first box is \( 6 \), second is \( b \), third is \( 10 \). But if we are to find \( b \), after substitution:
From \( 6^2 + b^2 = 10^2 \), \( b^2 = 100 - 36 = 64 \), so \( b = 8 \). But the initial substitution step (the boxes) should have \( 6 \), \( b \), \( 10 \). Wait, the problem says "Start by substituting the values you know and the value you don't know into the equation." So known values: \( a = 6 \), \( c = 10 \), unknown is \( b \). So substitute \( a = 6 \), \( c = 10 \), and \( b \) remains as \( b \). So the equation is \( 6^2 + b^2 = 10^2 \). So the boxes are filled with \( 6 \), \( b \), \( 10 \). But if we are to solve for \( b \), then:
After substitution: \( 36 + b^2 = 100 \)
Subtract 36: \( b^2 = 64 \)
Square root: \( b = 8 \)
But the first part is substitution. So the substitution is \( 6^2 + b^2 = 10^2 \), so the boxes are \( 6 \), \( b \), \( 10 \). But maybe the problem is just about the substitution step. So the answer for the substitution boxes is \( 6 \), \( b \), \( 10 \), and then solving gives \( b = 8 \).

Answer:

For the substitution: \( 6^2 + b^2 = 10^2 \) (so boxes: 6, b, 10). For \( b \): \( \boldsymbol{8} \)