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

Video Transcription

Problem 2. The base of a triangle increases by 50% and its height decreases by 1 third. What is the ratio of the area of the new triangle to that of the original one?  So to start, let's set variables for the original base and height of the triangle. So let's call them b and h respectively.  Then we know that the area of a triangle is the base times height divided by 2, so we know that the area of the original triangle was b h divided by 2.  Now let's look at the new triangle. We know that the base has increased by 50% and so that means that the new base is 150 over 100 times b, which is 3 halves of b.  We also know that the height decreased by 1 third. So from the full height, we have 1 third less, so the new height is 2 thirds of h. Now let's find the area of the new triangle.  So we have the new base times the new height divided by 2. So we have 3 halves b times 2 thirds h divided by 2.  The 3 halves and the 2 thirds cancel out and so the new area is just b h divided by 2. So now let's look for the ratio of the areas.  We know that the original triangle's area was b h over 2 and the new area is also b h over 2 and so their ratio is 1 to 1 and that tells us that the answer is b.

Video Summary

The problem involves calculating the ratio of the area of a modified triangle to its original form. Initially, the triangle's base is denoted as \(b\) and height as \(h\), with its area being \((b \cdot h)/2\). The base increases by 50% to become \(3/2 \cdot b\) and the height decreases by one-third to \(2/3 \cdot h\). The new area is calculated as \((3/2 \cdot b) \cdot (2/3 \cdot h)/2\), simplifying back to \((b \cdot h)/2\), the same as the original area. Thus, the ratio of the new area to the original is 1:1.