Grades 9-10 Video Solutions 2021-video 2021 9-10/21

Video Transcription

Problem number 21 states, an ant climbs from C to A on path C-A and descends from A to B on the stairs as shown in the diagram.  What is the ratio of the lengths of the ascending and descending paths? So here we have the path that the ant will travel, and here we have a triangle where  we know two-thirds of the angles, so let's start out by finding out the third angle. We can call that angle alpha, and we know that the sum of all the angles of a triangle  must be equal to 180 degrees. So alpha plus 75 plus 60 must be equal to 180. Subtracting 75 and 60 from both sides, we get the value of alpha, which is 45 degrees.  So perfect, now let's analyze how long the ascending path is. Let's split this triangle in half.  This would mean that the top angle turns into 45 degrees and 30 degrees, and this would make the angle at the bottom 90 degrees. So now let's consider this triangle.  Let's call its altitude x. We know that this triangle is a 30-60-90 triangle, so we know that the hypotenuse of the triangle is equal to 2 divided by square root 3 times x.  If you don't know how we got this, consider this 30-60-90 triangle. If the angle opposite the 30 degrees is x, then the hypotenuse will be 3x and the other side will be root 3x.  Divide everything by root 3, and we get 2x over root 3 for the hypotenuse. Now let's consider the descending path.  Here we can see it is a 45-45-90 isosceles triangle, so we know it's the two edges that are opposite the congruent angles have the same lengths, so we know that they're both x.  If we look at the staircase, we can see that the sum of the tops of the steps will add up to the base of the triangle, and the sums of the sides of the steps will add up to the  heights of the triangle. So both of them are x, which means the total length of the descending path is 2x.  Now let's find the ratios of the lengths of the ascending path to that of the descending path.  So 2 over root 3x over 2x. The 2's and the x's cancel out, and we get 1 over root 3. Let's get rid of the radical in the denominator by multiplying this by root 3 over root 3,  and we get root 3 over 3 as our final answer. So the question asks us, what is the ratio of the lengths of the ascending and descending paths? The answer is root 3 over 3, letter E.

Video Summary

The problem involves finding the ratio of the lengths of an ant's ascending and descending paths on a given triangle diagram. By calculating the angles and analyzing the triangle's properties, it is established that the ascending path involves a 30-60-90 triangle with a hypotenuse expressed as \( \frac{2}{\sqrt{3}}x \), while the descending path forms a 45-45-90 isosceles triangle, leading to a total length of \( 2x \) for the descending path. Simplifying the ratio of these lengths, the final answer is \(\frac{\sqrt{3}}{3}\).

Keywords

triangle diagram

ratio of lengths

30-60-90 triangle

45-45-90 triangle

path calculation