right scalene triangle problem with two right triangles in it help

V

Vol

Junior Member
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I just can't solve this one. I tried everything. None of the identities worked because there seems to be missing information.

Given: A triangle ABC from left to right, clockwise. Angles A and C are unknown, but angle B = 90. Sides a-b and b-c are also unknown. But from apex B a line is drawn straight down to point x at the base a-c. This vertical line b-x is perpendicular to base a-c. Thus the triangle is cut in half and form two right triangles within the larger right triangle. The length of a-x = 3. The length of x-c = 10. So, side a-c = 3 + 10 = 13. None of the others sides are known. Find the height of b-x. The answer choices are: a) 30 b) sqrt (30) c) 13 d) sqrt (109). Thanks for your help :giggle:
 
Vol said:
I just can't solve this one. I tried everything. None of the identities worked because there seems to be missing information.

Given: A triangle ABC from left to right, clockwise. Angles A and C are unknown, but angle B = 90. Sides a-b and b-c are also unknown. But from apex B a line is drawn straight down to point x at the base a-c. This vertical line b-x is perpendicular to base a-c. Thus the triangle is cut in half and form two right triangles within the larger right triangle. The length of a-x = 3. The length of x-c = 10. So, side a-c = 3 + 10 = 13. None of the others sides are known. Find the height of b-x. The answer choices are: a) 30 b) sqrt (30) c) 13 d) sqrt (109). Thanks for your help :giggle:
Click to expand...
Your terminology is non-standard, at least in my world; you use capitalization inconsistently, and write segments as if they were subtractions. I would say it this way:

Given: A triangle ABC, with side AC horizontal. Angles A and C are unknown, but angle B = 90. Sides AB and BC are also unknown. But from apex B a line is drawn vertically to point X at the base AC. This vertical line BX is perpendicular to base AC. Thus the triangle is cut in half and forms two right triangles within the larger right triangle. The length of AX is 3. The length of XC is 10. So, side AC = 3 + 10 = 13. None of the others sides are known. Find the height BX.​

Do you see any similar triangles? Look for them, and they will be the key.

I'd be interested to see an image of the problem as given to you, so we can see if your notation is just what you are taught in your culture or your class.
 
I am attaching a drawing (as a public service :)):
 

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There is a further, glaring problem with the question even after it has been correctly re-written by @Dr.Peterson ...
Dr.Peterson said:
Given: A triangle ABC, with side AC horizontal. Angles A and C are unknown, but angle B = 90. Sides AB and BC are also unknown. But from apex B a line is drawn vertically to point X at the base AC. This vertical line BX is perpendicular to base AC. Thus the triangle is cut in half into two and forms two right triangles within the larger right triangle. The length of AX is 3. The length of XC is 10. So, side AC = 3 + 10 = 13. None of the others sides are known. Find the height BX.​
Click to expand...

However, it can also be solved algebraically quite simply by using Pythagoras' Theorem & Simultaneous equations.

Borrowing @blamocur's excellent, wee sketch...

Triangles.png​

                                 \(\displaystyle \pmb {a^2 + c^2 = 169}\)
                                             \(\displaystyle \pmb {h^2 + 100 = a^2}\)
                                      and
                                 \(\displaystyle \pmb {h^2 + 9 = c^2}\)

Three equations & three variables. Simples. Reduced Aleksandr Orlov.jpg

Hope that helps. ☺️
 
Last edited:
Vol said:
Find the height of b-x
Click to expand...
Following the figure in response #3, we need to solve for only one unknown - length of BX.

As pointed out in response #4,

(BX)/(XC) = (AX)/(BX) → (BX) = \(\displaystyle \sqrt{(AX * XC)} \)
 
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