Math Olympiad Practice

Two rectangles of 5 by 1 and 3 by 1 are in the triangle ABC. What is the height of the triangle upon side BC? #math #maths #mathematics #shorts

Adapted from Math Kangaroo 2022, student level Q24

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[maxhagenauer]

I shifted it to make it a right triangle with a right angle on the bottom left. We know the base is 5 + (5 - 3) = 7. We also have 2 right triangles on the right that have legs of 1 and 2 so the hypotenuse for them is sqrt(5). Then we have the height of the rectangles on the left both with a height of 1. Call the height of the top triangle x and the hypotenuse for it y.

x^2 + 3^2 = y^2

(x + 1 + 1)^2 + 7^2 = (y +sqrt(5) + sqrt(5))^2

x = 1.5

1 + 1 + 1.5 = 3.5

[GoogleAccount-pict]

Simpler solution:

The sides of triangles are lines, so the change in base relative to height will be linear.
Adding 1 to height changes base from 5 to 3
Adding 1 more makes it 1
Adding another half makes it 0, aka the tip of the triangle, so we needed to add 1.5 more and there was already 2 so it's 3.5.

[dimpufrn]

I solved this but with a different method...

I noticed that the areas to the left and right of the small rectangle can be merged into a triangle. This triangle, let's call it triangle A, has a given height of 1 and base of (5 - 3) = 2.

I also noticed that the areas to the left and right of the large triangle can also be merged into a triangle, let's call it triangle B.

Since triangle A and B share the same angles and same height length upon the similar base, these two triangles are congruent to each other. Thus triangle B also has a base of 2.

We can now calculate the base of the large triangle, which is the length of the large rectangle + the base of triangle B = 5 + 2 = 7.

Since the large triangle shares the same angles as triangle A (or B), they are similar.
Thus the height of the large triangle can be calculated as follows:

1/2 = h/7
2h = 7
h = 3.5

[popogast]

Shift A to the left so that in B there's a right angle. Attach the inner rectangles to side AB. Side CA now has slope 1:2. Thus h is half of 3 (3:2 = 1.5).

[oussamajaber]

You forgot to say "And that's the answer."

[batchrocketproject]

This is a wonderful problem. Inspection of either rectangle shows the fundamental triangle (formed by joining the areas beside the rectangle) has height to width ration of 1/2. Since the top of the 5 rectangle forms the base of a triangle with base 5, it's height must be 2.5, and we know that line is 1 above the outer triangle's base. Giving 2.3+1 = 3.5. And that's the answer!.

[bruinjim]

Just look at the sequence. For every unit up, the width decreases by 2. So the base is 7. The height is 7/2 or 3.5.

[paull]

In case anyone wondered, the bottom side of the triangle is 7 long and the area is 12.25.

[fullfungo]

I used a different, more heuristic method.

The bases are all parallel and bounded by the sides of a triangle (which are straight lines). Therefore, the length of any given base is linearly proportional to the distance from the “top” point (because that’s precisely the relation between “linear” quantities and geometric “lines”).

So every step 1 unit from the “top” adds 2 units to the size of a given line parallel to the triangle’s base.
From this observation we can conclude two things:
1. The base of the triangle is 7.
2. H*2 = the base.

So H=3.5

[YourPalAlRetroGamer]

To solve for the base of the big triangle: Height of the small triangle is 1.5 and its base is 3. The height of the big triangle is 3.5, and we'll let x be its base. So, we have, 1.5/3 = 3.5/x. Simply the left hand side: 1/2 = 3.5/x. Cross multiply to get x= 2 X 3.5 = 7.

[varu]

More simpler you can do this using coordinate geometry,
The points where the side of triangle touches are (3, 2) and (5, 1).
Find the equation of line using these points i.e., *x+2y-7=0* substitute x=0 then you will get y=3.5. And that's the answer.

[mittelwelle__khz]

My method: the decrease in width is 2 units per 1 unit increase in height. So to reduce from the top side of the upper inscribed rectangle to zero (the top of the triangle) another 1.5 units increase in height are necessary. Plus the 2 units of height for the two inscribed rectangles makes 3.5 units.

But what do I know, I'm only an engineer, not a mathematician.

[cmuller]

3.5

If we call the small horizontal parts x (left) and y (right) we have:

3+x+y=5 (difference of size off the 2 rectangles) so x+y=2

(Edit : my "h" is the total height, not the height of the small top triangle of the explanation)
h.x+h.y=5+x+y (considering the big triangle with a vertical line of length h splitting it into 2 parts similar to the small triangles, this is the length of the base of the big triangle)
So 2.h=7
h=3.5

[farrier]

Assume the rectangles are placed symmetrically, then the problem reduces to a 3x2 rectangle within a 45 deg right triangle. The apex triangle above the rectangle can then be split vertically from the apex, to form two right triangles with both sides 1.5. Thus, the side of 1.5 plus the height, 2, of the rectangle is the answer. It was pure geometry. No algebra needed.

[TomKaren]

The portion of the base of each of the triangles outside the rectangles is 2. Since the height of the rectangles is, of course, consistent at 1, the base lengthens by 2 when the height increases by 1. The largest base is easily seen to be 7, therefore the height must be 3.5 .

[Broadsmile]

the sides converge with a rate of 2 horizontal distance per 1 vertical distance. Or in other words, the lines come closer by 1 horizontal distance over the course of 0.5 vertical distance. Multiply the "remaining" horizontal distance by this rate of 0.5, and you get 1.5 + the height up to that "remaining" part 1.5 + 2 = 3.5

[edehans]

Happy 2023. Thanks for all your puzzles! It's not difficult to see that the distance between the two sides of the triangle must diminish linearly with height. It diminishes by 2 when going up 1 from 5 to 3. So, it's 0 (curves meet) another 3/2=1.5 higher. We were already at height 2, so 3.5 in total.

[Waldlaeufer]

Let's move all the rectangles to one side and we'll get m = 1/2. Therefore, base b = 7 and height h = 3.5

[michaeljohnmagistrado]

It's amazing how many geometry questions like this is solvable just using ratios.

[RomshaAwasthiA]

a trick question having an elegant solution

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