Question

geometry
concept quiz n
date: 11/25/25
c#13 in isosceles triangle hat, solve for x, and use it to find the perimeter of the triangle.
c#14 complete the proof to show that the two triangles are congruent.
given: a is the midpoint of \\( \overline { p e } \\)
a is the midpoint of \\( \overline { l n } \\)
prove: \\( \triangle p l a \cong \triangle e n a \\)

Explanation:

Step1: Use the property of isosceles triangle

In an isosceles triangle \(HAT\), the two equal - side lengths are given by \(5x + 19\) and \(7x-11\). Since the sides \(AH\) and \(AT\) are equal (marked with the same tick - mark), we set up the equation \(5x + 19=7x - 11\).

Step2: Solve the equation for \(x\)

Subtract \(5x\) from both sides of the equation:
\(5x+19 - 5x=7x - 11-5x\)
\(19 = 2x-11\)
Add \(11\) to both sides:
\(19 + 11=2x-11 + 11\)
\(30 = 2x\)
Divide both sides by \(2\):
\(x=\frac{30}{2}=15\)

Step3: Find the lengths of \(AH\) and \(AT\)

Substitute \(x = 15\) into \(5x + 19\):
\(5\times15+19=75 + 19=94\)
Substitute \(x = 15\) into \(7x-11\):
\(7\times15-11=105 - 11 = 94\)

Step4: Calculate the perimeter of \(\triangle HAT\)

The perimeter \(P\) of a triangle is \(P=a + b + c\), where \(a,b,c\) are the side lengths of the triangle. Here \(a = 94\), \(b = 94\), \(c = 130\)
\(P=94+94 + 130=318\)

Answer:

\(x = 15\)
Perimeter \(=318\)