Question

Please solve q 100?

Answer 1

2 (I'm probably wrong sorry in advance)

Explanation:

Try drawing it out. For Geometry, it is very important to be able to draw well. Drawing out the question can help you understand and see it more clearly.

For this, you would draw a right triangle `PQR` and make sure to label it clearly. Read the question for the information.

• It tells you that `PQ` is the hypotenuse.
• And then there are the two points, `S and T`.

These trisect the hypotenuse, meaning they should split the hypotenuse into three equal sections/line segments: `PS, ST, TQ` (or something like that.) As for the actual solution, I'm not sure but I think it's `2`? But that seems too simple so.... I'm sorry!! I know it's not an answer but I hope it helps. Again, I apologize.

Answer 2

`sqrt{26}`, choice 1.

Explanation:

Let's call the legs `p=PR and q=QR.` We'll call `x=ST`. Then the hypotenuse `PQ=3x`.

The three triangles PRS, SRT and TRQ have a common altitude to `R` and equal bases, so the same area, one third the area of the big triangles.

Let's call `A` the area of PRS.

`3A = 1/2 pq`

`pq = 6A`

We'll use Archimedes Theorem relating area `A` to triangle sides, `a,b,c`

`16A^2 = 4a^2b^2-(c^2-a^2-b^2)^2`

RST: `quad 16A^2 = 4(7^2)(9^2)-(x^2-7^2-9^2)^2`

PRS: `quad 16A^2 = 4(7^2)(p^2)-(x^2-p^2-7^2)^2`

QRT: `quad 16A^2 = 4(9^2)(q^2)-(x^2-q^2-9^2)^2`

`pq = 6A`

`p^2 + q^2 = (3x)^2`

That's five equations in four unknowns, `A^2,p^2,q^2,x^2`

I'm just gonna feed it to Alpha. I'll write the variables in caps as the squares. This is better, it avoids negative roots.
`{16 A = 4×7^2×9^2 - (X - 7^2 - 9^2)^2 = 4×7^2 P - (X - P - 7^2)^2 = 4×9^2 Q - (X - Q - 9^2)^2, P + Q = 9 X, P Q = 36 A}`

Alpha reports

`A^2 = 1265/4, p^2 = 69, q^2 = 165, x^2 = 26`

`x=sqrt{26}` is choice 1.

Right triangle Pythagorean Theorem: `9 times 26 = 69 + 165 quad sqrt`

FYI `arctan (sqrt{165/69}) approx 57^circ`. We don't care about the approximate angle because we have all the trig functions, including the tangent, aka slope.

That works.

EDIT: I thought there were four solutions but I had left an equation out.