Can You Find the Area of this Semicircle? | Step-by-Step Explanation

[martinmarc-olivier]

Very clear explanations.
A faster way to reach the split k, 4k, is to notice that both triangles have the same height, therefore their bases must have the same ratio as their areas, that ratio being 28/7 = 4.

[andreadanieli]

I love thinking in an alternative way!

Instead of introducing the intersecting chords theorem I considered proportionality of green and purple triangles.
n : h = h : m ---> h^2 = m * n ---> h^2 = 4 * m^2 ---> h = 2m
Since m * h = 14 ---> m * 2m = 14 ---> m^2 = 7 ---> m = sqrt(7)
Since n = 4m ---> n = 4 * sqrt(7)
So we can obtain the diameter of the circle D = m + n = 5 * sqrt(7)
Finally, area of the semi-circle is A = pi * D^2 / 8 = pi * 25 * 7 /8 = pi * 175/8 ; - )

[skoijlg]

It is clear that one more equation is needed to find the radius. We get from the similarity of right triangles n/h=h/m. mn= h^2. Multiply the first two equations, we get h^4=56*14. Further it is clear. The relationship between the height of a right triangle and the segments of the hypotenuse is studied in the school curriculum. The height is the average proportional between the projections of the legs on the hypotenuse. You probably have a different program from ours.

[johnbrennan]

Very nice method. Another way to get h in terms of k instead of intersecting chord theorem is to use the theorem Pythagorous on the three triangles.

[spafon]

I haven't checked the solution yet, but I would use similar triangles. The small triangle has small side a. Because the area of the larger triangle is 4x that of the small triangle then the base and height of the larger triangle are each 2x of those of the small triangle, hence the small side (height) of the larger triangle is 2a. Thus the small triangle has base a and height 2a. Thus its area is a^2=7 and thus a=sqrt(7). Since the triangles are similar, and the height/base of the larger triangle = the base/height of the small triangle, which is 1/2. Thus the base of the larger triangle is twice the height of that triangle 2a, and thus is 4a. The diameter of the semicircle is thus 5a, the radius is 2.5a, thus 2.5*sqrt(7). Hopefully. Now I'll check the solution.

[boanergesct]

The sum triangle of the two represented is a right triangle since it is inscribed in a semicircle (diameter 2R). Both triangles are similar since they have the three equal angles; their bases (“b” and “B”) add up to 2R and their lengths are in the same ratio as the areas (28/7=4), since they share the same height “h” → 4b=B → 2R=b+ B=b+4b=5b → 2R=5b → b=R2/5 → B=4b=R8/5. The similarity between both triangles allows us to write the following equality: b/h=h/B → (R2/5)/h=h/(8R/5) → h=R4/5.
With the obtained values we can write: bh/2=7 → (R2/5)(R4/5)/2=7 → R²8/25=2x7=14 → R²=14x25/8 → Area of the semicircle = πR²/2 = π14x25/2x8 = π175/8 = 68.7223

[highplayz]

I done it mentally by using similarity
Theorem used :
If in a right triangle
Perpendicular drawn from 90° to hypotenuse then 2 triangles formed are similar to each other and to the whole triangle

[kennethstevenson]

Since both triangles have a common height and one area is four times the other then the green base is 4 times the purple base. The triangles are similar, producing the proportions h/k = 4k/h. Cross multiply you get h^2 = 4k^2, simplify h=2k. Substituting green Area formula 28=1/2 x 4k x 2k ; 28=4k^2 ; 7 = k^2 and finally k= square root of seven. The radius is 5k/2 . The rest is the same without using the Intersecting Chord Theorem, which was a good thing to review. Math done correctly produces consistent results!

[ramaprasadghosh]

Left part of Diameter/ Its Right part
= 7/28 = 1/4 = D_L/ D_R say
Hereby D_L = D/5, D_R = 4D/5
Congruence triangle property gives
H /D_R = D_L/ H
or H^2 = D_L * D_R = 4 *D^2/ 5^2
or H = 2*D/5
Now H*D/2 = 7 + 28 = 35
so D^2/5 = 35 i.e D^2 = 175
Hereby area of semi circle
π*D^2/8 = 175*π/8

[misterenter-izrz]

2a^2=14, 8a^2=56, a^2=7, 5a is the diameter, 5a/2 is the radius, therefore the answer is pi=68.7 approximately.

[gikasmith]

Nice. Although I think noticing that the entire triangle - green plus purple - is a right triangle (inscribed into a semicircle, inscribed angles property) would make the solution even easier. The two are therefore similar, so the height is the geometric mean of the projections - and voila.

[theoyanto]

What can I say, superlatives fail me, but I'll try... Exceptional example, above and beyond the call of duty, you exceeded even your own high standard, what a fantastic example, there was no way I could have completed this without your video. Thanks again 👍🏻

[brucecarter]

the 2 triangles are similar because the 2 chords make a 90 degree angle. the area of the larger is 4x that of the smaller, so the sides are 2x longer. if the short leg of the 7 triangle is x, then the short leg of the 28 triangle is 2x (also the long leg of the 7 triangle so the long leg of the 28 triangle is 4x)
1/2*x*2x=7
x=sqrt7
D=5x=5sqrt7

[Khalid]

Also we can do it by adding the A1and A2 which give us 35 and find h since it is 90° tringle.

[flexeos]

when you have m and n =f(h) and 2r=a+b, you can also use directly r^2 = ((n-m)/2)^2 + h^2

[holyshit]

Draw radius from center to vertex outside diameter
Let x be the base of purple triangle
Calculate h in terms of x from formula of area of this purple right trangle
From Pythagorean theorem in right triangle with sides h, R-x, R
we calculate R in terms of x
From formula of area of right triangle coloured on green we calculate R
once we have R, it is easy to calculate area of semicircle

[govindashit]

Another great solution.
It's very helfful for all students.
Thanks dear sir.

[basavarajchikkamath]

Very good problem, nice solution. Thank you sir. 👌🙏🙏

[crabbyengineer]

The initial problem statement should have included the assumption that both green and purple triangles are right triangles.

[dreael]

You also can use the Thales's theorem, i.e. the bottom angle is also 90 degrees. With the angles you can show that the green and purple triagle are similar, even the resulting triangle of adding both triangles is similar. So because of that, h must be the geometric mean of m and n.