Question

3-109. the area of the triangle below is 25 square units. find the value of h. then find the perimeter of the entire triangle. show all work.

Explanation:

Step1: Find the base length

The base of the triangle is the sum of the two segments, so \( b = 6 + 4 = 10 \) units.

Step2: Use the area formula to find \( h \)

The area formula for a triangle is \( A=\frac{1}{2}bh \). We know \( A = 25 \) and \( b = 10 \). Substitute these values into the formula:

$$ 25=\frac{1}{2}(10)h $$

Simplify the right - hand side: \( \frac{1}{2}(10)h=5h \). Then we have the equation \( 5h = 25 \). Divide both sides by 5: \( h=\frac{25}{5}=5 \) units.

Step3: Find the lengths of the other two sides

Using the Pythagorean theorem \( c=\sqrt{a^{2}+h^{2}} \) for each right - triangle formed by the height \( h \).
For the segment of length 6: Let the length of the side be \( l_1 \). Then \( l_1=\sqrt{6^{2}+5^{2}}=\sqrt{36 + 25}=\sqrt{61}\approx7.81 \) units.
For the segment of length 4: Let the length of the side be \( l_2 \). Then \( l_2=\sqrt{4^{2}+5^{2}}=\sqrt{16 + 25}=\sqrt{41}\approx6.40 \) units.

Step4: Calculate the perimeter

The perimeter \( P \) of the triangle is the sum of all three sides. The three sides are \( \sqrt{61} \), \( \sqrt{41} \), and \( 10 \).

$$ P=\sqrt{61}+\sqrt{41}+10\approx7.81 + 6.40+10=24.21 $$

(If we want an exact form, the perimeter is \( 10+\sqrt{41}+\sqrt{61} \))

Answer:

The value of \( h \) is 5 units. The perimeter of the triangle is \( 10+\sqrt{41}+\sqrt{61}\approx24.21 \) units (or in exact form \( 10 + \sqrt{41}+\sqrt{61} \) units).