To find the height of an isosceles triangle where the base is 15 inches and each of the two congruent sides is 12 inches, we need to apply the Pythagorean theorem.
Here's a step-by-step approach:
1. Identify Key Elements:
- Base of the isosceles triangle: 15 inches
- Length of each congruent side: 12 inches
2. Divide the Base:
- When we draw a height from the top vertex perpendicular to the base, it bisects the base into two equal segments.
- Each segment of the base is: [tex]\( \frac{15}{2} = 7.5 \)[/tex] inches
3. Form Two Right Triangles:
- By drawing the height, we now have two right-angle triangles, each with:
- One leg = 7.5 inches (half of the base)
- Hypotenuse = 12 inches (congruent side)
4. Use the Pythagorean Theorem:
- The Pythagorean theorem states: [tex]\( a^2 + b^2 = c^2 \)[/tex]
- Here:
- [tex]\( c \)[/tex] (hypotenuse) = 12 inches
- [tex]\( b \)[/tex] (one leg along the height) = 7.5 inches
- [tex]\( a \)[/tex] (the other leg - the height) is what we need to find
5. Set Up the Equation:
[tex]\[
a^2 + 7.5^2 = 12^2
\][/tex]
6. Calculate:
[tex]\[
a^2 + 56.25 = 144
\][/tex]
[tex]\[
a^2 = 144 - 56.25
\][/tex]
[tex]\[
a^2 = 87.75
\][/tex]
[tex]\[
a = \sqrt{87.75}
\][/tex]
7. Find the Height:
[tex]\[
a = 9.367496997597597
\][/tex]
8. Round to the Nearest Tenth:
- The height rounded to the nearest tenth is [tex]\(9.4\)[/tex] inches.
So, the correct answer is:
D. 9.4 inches
