SAT Math : How to find if right triangles are similar
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Example Questions
Example Question #1 :Right Triangles
In the figure above, line segmentsDCandABare parallel. What is the perimeter of quadrilateralABCD?
95
80
90
75
85
85
BecauseDCandABare parallel, this means that anglesCDBandABDare equal. When two parallel lines are cut by a transversal line, alternate interior angles (such asCDBandABD) are congruent.
Now, we can show that trianglesABDandBDCare similar. BothABDandBDCare right triangles. This means that they have one angle that is the same—their right angle. Also, we just established that anglesCDBandABDare congruent. By the angle-angle similarity theorem, if two triangles have two angles that are congruent, they are similar. Thus trianglesABDandBDCare similar triangles.
We can use the similarity between trianglesABDandBDCto find the lengths ofBCandCD. The length ofBCis proportional to the length ofAD, and the length ofCDis proportional to the length ofDB, because these sides correspond.
We don’t know the length ofDB, but we can find it using the Pythagorean Theorem. Leta,b, andcrepresent the lengths ofAD,AB, andBDrespectively. According to the Pythagorean Theorem:
a2+b2=c2
152+ 202=c2
625 =c2
c= 25
The length ofBDis 25.
We now have what we need to find the perimeter of the quadrilateral.
Perimeter = sum of the lengths ofAB,BC,CD, andDA.
Perimeter = 20 + 18.75 + 31.25 + 15 = 85
The answer is 85.
Example Question #114 :Sat Mathematics
A traffic light hangstfeet from the ground, over a street. A man standing the shadow of the traffic light ishfeet tall, and his shadow issfeet long. How far is the man standing from the spot on the street directly under the traffic light?
We can set this problem up like a set of similar triangles.
The first triangle is created by the three points: The Traffic light, the spot beneath the traffic light, and the spot where the man is standing (which is also the spot where the traffic light's shadow is).
The height of this Triangle is "T" as given in the question, and its base is the part that we are asked to solve for.
The second triangle is created by the top of the man's head, his feet, and the end of his shadow.
The height of this Triangle is "h" as given in the questions, and the base is "s".
We set up a proportion:
where X is the distance we are asked to find. Simply cross-multiply to solve.
Example Question #112 :Sat Mathematics
An meteor crashed in the desert and created an oblong shaped crater. Scientists want to find the width
To calculate the width of the crater, use the given information to establish that the image draws similar triangles. When triangles that have corresponding angles and a ratio to their side lengths they are considered to be similar triangles.
Identify the known information.
therefore,
and the bases of the triangles are parallel.
Also,
Set up the side ratios for this particular problem.
Looking at the only full ratio that is given, the scalar multiplier can be found.
Therefore, to find the width of the crater
Example Question #116 :Sat Mathematics
In the given diagram,
By the Pythagorean Theorem,
Set
Take the positive square root of both sides:
Set
A right triangle has as its area half the product of the length of its legs, so the area of
To the nearest whole number, this rounds to 35.
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