SAT original mathematics module 1 question 17

We are looking for 'x', which is the distance from 'A' to 'E' in feet.

What do we know?

We can find out:
a) from "... by AB, EB, BD, and CD... to be 1800..., 1400..., 700..., and 800..., respectively." That AB = 1800, EB = 1400, BD = 700, and CD = 800.

b) from "Segment AC and DE intersect at B,..." That ^(angle)ABE = ^CBD (vertically opposite angles).

c) lastly, from "... and ^AEB and ^CDB have the same measure." That ^AEB = ^CDB.

To find 'x' we need to apply the knowledge of similar triangles

Now, what are similar triangles?
Similar triangles are triangles that are simply similar, it's only their size that differentiates them. Their angles are the same.
Unlike congruent triangles that have the same angles and length, similar triangles have the same angles, but may have different lengths, which would be in the same ratio to eachother

How do we identify similar triangles?

i) AA (angle-angle); this is when two angles are congruent (the same) in the given triangles
It identifies similar triangles because the last angle would obviously be the same also, and using sine law, their lengths would be in the same ratio as well.

ii) SAS (side-angle-side); this is when one angle is congruent in the given triangles; and two sides of the triangle are in the same ratio as their corresponding sides of the other triangles
It identifies similar triangles because the ratio of the given sides would provide the necessary requirements for two angles to be congruent, allowing the triangles to be similar

iii) SSS (side-side-side); this is when there is a common ratio between the three sides of a triangle and it's corresponding side on the other triangles
It identifies similar triangles because the ratio of the given sides would provide the necessary requirements in line with sine rule for all the angles to be the same, allowing the triangles to be similar

NOTE: two triangles can be formed from the give shape (🔺BAE and 🔺BCD) and how do we know they are congruent?
We were not given all the sides of the two triangles, so it can't be SSS; and we were only given a pair of corresponding side, so it can't be SAS; but we were give two pairs of angles that are congruent (^ABE = ^CBD; and ^AEB = ^CDB), so AA can prove that they are similar triangles.

Let's solve:
Before solving any similar triangle questions you must first of all determine the ratio of the triangle. Let🔺BAE be TM and🔺BCD be TN. The ratio could be gotten using the angles, but in this case they were not given, so we will use any two side that has it's corresponding side given.
Line EB and BD correspond, so we can use them
Line EB is on 🔺BAE, while line BD is on 🔺BCD; and we are looking for 🔺BAE/🔺BCD, so we do line EB/ line BD
EB = 1400, BD = 700
∴EB/BD = 1400/700 = 2
That means all the lines on 🔺BAE are two times that on 🔺BCD

We are looking for 'x' which is line AE
It's corresponding side is line CD, which was given to be 800
∴AE = 2×800 = 1600;
x = 1600.

There're no options in this question, but this is how you would answer this question appropriately.

You simply shade number 1 on the first column, followed by number 6 on the second column, then zeros on the remaining two columns
i.e. 1600
Then you write '1600' in the space provided at the top of the column on the first row

Kabadosh

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