Grades 9-10 Video Solutions 2025-2025_9-10

Video Transcription

Problem 7. Alex folds a square in half along a diagonal to make a triangle. Then he folds the paper again so that one of the short edges of the triangle lies on top of the long edge  of this triangle, making the smaller triangle AXC as shown. What is the measure of angle AXC? So this angle right here. So first we know that from triangle ABC we have that this  angle CAB is the same as the angle ACB and they're both 45. This is because AB and BC are both side lengths of a square so they have the same length, and that means that  ABC is an isosceles triangle so these two angles are the same, and we also know from the square that the angle at B here is 90. Also we can figure out that ACX and XCB, so  these two angles here, have the same measure. That's because when we fold over the line XC, those two angles will overlap and so it must be that they're the same measure.  Now ACX and XCB together make the angle ACB which we already knew was 45, and so ACX has to be half of that angle and so it has measure 22.5.  Now if we go to triangle AXC, we know that the sum of all three angles has to be 180, and we know that CAX is 45 and ACX is 22.5 as we previously found, and so that means that AXC  here has a measure of 112.5, and so our answer is B.

Video Summary

In the problem, Alex folds a square into a triangle and then refolds to create another triangle. The goal is to determine the measure of angle AXC. Initially, angles within the square-turned-isosceles triangle ABC are analyzed, establishing that angles CAB and ACB are each 45 degrees. Through folding, angles ACX and XCB are found to be equal and together add up to 45 degrees, making each 22.5 degrees. In triangle AXC, with angles CAX at 45 degrees and ACX at 22.5 degrees, angle AXC is calculated to be 112.5 degrees. Thus, the measure of angle AXC is 112.5 degrees.

Keywords

square folding

triangle angles

isosceles triangle

angle calculation

geometry problem