Find the area of each isosceles triangle . Show work ! ### Geometry Problem: Finding the Height of an Isosceles Triangle#### Problem StatementIn an isosceles triangle, the lengths of the two equal sides are 75 feet each, and the length of the base is 42 feet. Determine the height (h) of the triangle.#### DiagramBelow is a diagram of the isosceles triangle:``` /|\ / | \ / | \ / | \ /____|____\ h```The diagram includes:- Two equal side lengths labeled as 75 ft.- A base with a length of 42 ft.- An altitude (height) from the top vertex perpendicular to the base, labeled as h, which forms two right triangles.#### ExplanationConsidering the symmetry of the isosceles triangle, the altitude h splits the base into two equal segments of 21 feet each. This creates two right triangles where:- The hypotenuse is 75 ft.- One leg is 21 ft (half of the base).- The other leg is the height (h) of the triangle.Using the Pythagorean theorem in one of the right triangles:\[ a^2 + b^2 = c^2 \]Substitute the known values:\[ h^2 + 21^2 = 75^2 \]Calculate the height (h):\[ h^2 + 441 = 5625 \]\[ h^2 = 5625 - 441 \]\[ h^2 = 5184 \]\[ h = \sqrt{5184} \]\[ h = 72 \]Thus, the height (h) of the triangle is 72 feet.

An isoceles triangle is a triangle in which 2 sides are equal. The third side of such a triangle…

Answered: 75 ft h 75 ft 42 ft

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

ChapterP: Preliminary Concepts

SectionP.CT: Test

Problem 1CT

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Find the area of each isosceles triangle . Show work !

### Geometry Problem: Finding the Height of an Isosceles Triangle

#### Problem Statement
In an isosceles triangle, the lengths of the two equal sides are 75 feet each, and the length of the base is 42 feet. Determine the height (h) of the triangle.

#### Diagram
Below is a diagram of the isosceles triangle:

```
/|\
/ | \
/ | \
/ | \
/____|____\
h
```

The diagram includes:
- Two equal side lengths labeled as 75 ft.
- A base with a length of 42 ft.
- An altitude (height) from the top vertex perpendicular to the base, labeled as h, which forms two right triangles.

#### Explanation
Considering the symmetry of the isosceles triangle, the altitude h splits the base into two equal segments of 21 feet each. This creates two right triangles where:

- The hypotenuse is 75 ft.
- One leg is 21 ft (half of the base).
- The other leg is the height (h) of the triangle.

Using the Pythagorean theorem in one of the right triangles:

\[ a^2 + b^2 = c^2 \]

Substitute the known values:
\[ h^2 + 21^2 = 75^2 \]

Calculate the height (h):
\[ h^2 + 441 = 5625 \]
\[ h^2 = 5625 - 441 \]
\[ h^2 = 5184 \]
\[ h = \sqrt{5184} \]
\[ h = 72 \]

Thus, the height (h) of the triangle is 72 feet.

Polygon with three sides, three angles, and three vertices. Based on the properties of each side, the types of triangles are scalene (triangle with three three different lengths and three different angles), isosceles (angle with two equal sides and two equal angles), and equilateral (three equal sides and three angles of 60°). The types of angles are acute (less than 90°); obtuse (greater than 90°); and right (90°).

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