Answer:
[tex]x=8[/tex]
Step-by-step explanation:
We know that this is an isosceles triangle from the given information. We are also given that its height, or altitude, is 8. The height/altitude to the base of an isosceles triangle is also the median to the base. A median is a line segment that bisects another line segment, or splits it into two equal parts. Therefore, we know that the vertical line segment in the diagram splits the base into two equal parts, each measuring [tex]\frac{x}{2}[/tex] units.
Now, we know that the altitude of a line segment is perpendicular to it, or forms a [tex]90[/tex]° angle with it. Therefore, the vertical segment divides the larger isosceles triangle into 2 congruent right triangles with legs of 8 and [tex]\frac{x}{2}[/tex] and hypotenuse of [tex]\sqrt{80}[/tex]. We can use the Pythagorean Theorem ([tex]a^{2} +b^{2} =c^{2}[/tex] where [tex]a[/tex] and [tex]b[/tex] are the legs and [tex]c[/tex] is the hypotenuse of the right triangle) to solve for [tex]x[/tex]. Substituting the values for [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] into the Pythagorean Theorem equation, we get:
[tex]a^{2} +b^{2}=c^{2}[/tex]
[tex](\frac{x}{2})^{2} +8^{2}=(\sqrt{80})^{2}[/tex] (Substitute values for [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex])
[tex]\frac{x^{2} }{4}+64=80[/tex] (Simplify)
[tex]\frac{x^{2} }{4}+64-64=80-64[/tex] (Subtract 64 from both sides)
[tex]\frac{x^{2} }{4} =16[/tex] (Simplify)
[tex]\frac{x^{2} }{4}*4=16*4[/tex] (Multiply both sides of the equation by 4)
[tex]x^{2} =64[/tex] (Simplify)
[tex]\sqrt{x^{2} }=\sqrt{64}[/tex] (Take the square root of both sides of the equation)
[tex]x=8,x=-8[/tex] (Simplify)
[tex]x=-8[/tex] is an extraneous solution because a triangle cannot have negative side lengths. Thus, our final answer is [tex]x=8[/tex]. Hope this helps!
