Geometry Similarity Solving Problems with Similar and Congruent Triangles
1 Answer
sankarankalyanam
Apr 27, 2018
#color(maroon)(x = 15 / 4, color(green)(y ~~ 9.8 " units"#
Explanation:
Since triangles ABC and DEF are similar, their corresponding sides are in the same ratio.
#x / 5 = 9 / 12 = y / b#
#color(maroon)(x = (5 * 9) / 12 = 15 / 4#
From Pythagoras theorem,
#y^2 = x^2 + 9 ^2 = (15/4)^2 + 9^2 = 95.0625#
#color(maroon)(y = sqrt (96.0625) ~~ 9.8 " units"#
Answer link
Related questions
Question #fe069
What are the differences between similar triangles and congruent triangles?
Are two isosceles triangles always similar? Are two equilateral triangles always similar? Give...
If two triangles are congruent, are they similar? Please explain why or why not.
In the figure given identify the congruent and/or similar triangles and find the value of x and y?
Find the value of x in the figure?
Question #7ef19
Question #3cf21
Why can't there be an axiom of congruency of triangles as A.S.S. similar to R.H.S.?
Given the figure determine the value of the unknown segment, #x#?
See all questions in Solving Problems with Similar and Congruent Triangles
Impact of this question
8920 views around the world
You can reuse this answer
Creative Commons License
If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.
More specifically, you’re going to see how to use the geometric mean to create proportions, which in turn help us solve for missing side lengths.
Let’s get started!
How are right triangles and the geometric mean related?
A right triangle has two acute angles and one 90° angle.
The two legs meet at a 90° angle, and the hypotenuse is the side opposite the right angle and is the longest side.
Right Triangle Diagram
The geometric mean of two positive numbers a and b is:
Geometric Mean of Two Numbers
And the geometric mean helps us find the altitude of a right triangle! In fact, the geometric mean, or mean proportionals, appears in two critical theorems on right triangles.
Geometric Mean Theorems
In a right triangle, if the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments, then the length of the altitude is the geometric mean of the lengths of the two segments.
Altitude Rule
Additionally, the length of each leg is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg, as ck-12 accurately states.
Leg Rule
But what do these theorems really mean?
They help us to create proportions for finding missing side lengths!
Let’s look at an example!
How To Solve Similar Right Triangles
In the figure below, we are being asked to find the altitude, using the geometric mean and the given lengths of two segments:
Using Similar Right Triangles
In the video below, you’ll learn how to deal with harder problems, including how to solve for the three different types of problems: