1. In triangle XYZ, A is the midpoint of XY, B is the midpoint of YZ, and C is the midpoint of XZ. Also, AY = 7, BZ = 8, and XZ = 18. What is the perimeter of triangle ABC? (SHOW WORK)2. What is y? (SHOW WORK) 2nd picture is the triangle.
Accepted Solution
A:
Answer:Part 1) The perimeter of triangle ABC is 24 unitsPart 2) [tex]y=97\°[/tex]Step-by-step explanation:Part 1) we know thatThe Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third sidesee the attached figure to better understand the problemsoApplying the midpoint theoremstep 1Find the value of BC[tex]BC=\frac{1}{2}XY[/tex][tex]XY=2AY[/tex] ---> because A is the midpointsubstitute the given value of AY[tex]XY=2(7)=14\ units[/tex][tex]BC=\frac{1}{2}(14)=7\ units[/tex]step 2Find the value of AC[tex]AC=\frac{1}{2}YZ[/tex][tex]YZ=2BZ[/tex] ---> because B is the midpointsubstitute the given value of BZ[tex]YZ=2(8)=16\ units[/tex][tex]AC=\frac{1}{2}(16)=8\ units[/tex]step 3Find the value of AB[tex]AB=\frac{1}{2}XZ[/tex]substitute the given value of XZ[tex]AB=\frac{1}{2}(18)=9\ units[/tex]step 4Find the perimeter of triangle ABC[tex]P=AB+BC+AC[/tex]substitute[tex]P=9+7+8=24\ units[/tex]Part 2) Find the measure of angle ystep 1Find the measure of angle zwe know thatThe sum of the interior angles in a triangle must be equal to 180 degreesso[tex]55\°+42\°+z=180\°[/tex]solve for z[tex]97\°+z=180\°[/tex][tex]z=180\°-97\°[/tex][tex]z=83\°[/tex]step 2Find the measure of angle ywe know that[tex]z+y=180\°[/tex] ----> by supplementary angles (form a linear pair)we have[tex]z=83\°[/tex]substitute[tex]83\°+y=180\°[/tex]solve for y[tex]y=180\°-83\°[/tex][tex]y=97\°[/tex]