1. ## Triangle (Show)

Anyone can show me the method?

1. In a triangle PQR, PQ=a, QR=a+d, RP=a+2d and $\displaystyle \angle PQR > 120^o$, where a and d are positive constant, show that $\displaystyle \frac{2}{3}a<d<a$

2. A man observed the top of a tower from the points A, B and C along a straight, horizontal road, where AB=a and BC=b. The angle of elevation of the top of the tower from A is $\displaystyle \alpha$ while the angle of elevation from B and C are both equal to $\displaystyle \beta$. If $\displaystyle \vartheta$ is the angle of elevation of the top of the tower from the point E wich is the mid-point of BC, show that

i)$\displaystyle \vartheta$ is the largest angle of elevation,
ii) $\displaystyle 4a(a+b)cot^2 \vartheta = (2a+b)^2 cot^2 \beta - b^2 cot^2 \alpha$

2. Hello cloud5
Originally Posted by cloud5
Anyone can show me the method?

1. In a triangle PQR, PQ=a, QR=a+d, RP=a+2d and $\displaystyle \angle PQR > 120^o$, where a and d are positive constant, show that $\displaystyle \frac{2}{3}a<d<a$

2. A man observed the top of a tower from the points A, B and C along a straight, horizontal road, where AB=a and BC=b. The angle of elevation of the top of the tower from A is $\displaystyle \alpha$ while the angle of elevation from B and C are both equal to $\displaystyle \beta$. If $\displaystyle \vartheta$ is the angle of elevation of the top of the tower from the point E wich is the mid-point of BC, show that

i)$\displaystyle \vartheta$ is the largest angle of elevation,
ii) $\displaystyle 4a(a+b)cot^2 \vartheta = (2a+b)^2 cot^2 \beta - b^2 cot^2 \alpha$
Here's the outline for number 1.

$\displaystyle 120^o< \angle Q <180^o \Rightarrow -\tfrac12>\cos Q >-1$
$\displaystyle \Rightarrow -\tfrac12>\frac{a^2+(a+d)^2-(a+2d)^2}{2a(a+d)}>-1$, using the Cosine Rule.
Expand and simplify:
$\displaystyle \Rightarrow ...$
Factorise the numerator, and cancel.
$\displaystyle ...$
Reverse the direction of the inequality (multiply through by -1).
$\displaystyle ...$
$\displaystyle \Rightarrow \tfrac23a<d<a$
Can you complete it?

3. Hello again cloud5
Originally Posted by cloud5
...2. A man observed the top of a tower from the points A, B and C along a straight, horizontal road, where AB=a and BC=b. The angle of elevation of the top of the tower from A is $\displaystyle \alpha$ while the angle of elevation from B and C are both equal to $\displaystyle \beta$. If $\displaystyle \vartheta$ is the angle of elevation of the top of the tower from the point E wich is the mid-point of BC, show that

i)$\displaystyle \vartheta$ is the largest angle of elevation,
ii) $\displaystyle 4a(a+b)cot^2 \vartheta = (2a+b)^2 cot^2 \beta - b^2 cot^2 \alpha$
... and here's the outline for number 2.

Suppose that $\displaystyle P$ is the foot and $\displaystyle T$ the top of the tower, and that $\displaystyle TP = h$. Then we have:
$\displaystyle \angle TAP = \alpha$

$\displaystyle \angle TBP = \angle TCP = \beta$

$\displaystyle \angle TEP = \theta$

$\displaystyle AB = a, BE = EC = \tfrac12b$
For part (i): $\displaystyle PB = h\cot\beta = PC$
$\displaystyle \Rightarrow \triangle BPC$ is isosceles

$\displaystyle \Rightarrow PE \perp BC$

$\displaystyle \Rightarrow PE$ is the shortest distance from the foot of the tower to $\displaystyle BC$.

So?
(ii) $\displaystyle PE^2 = h^2\cot^2\theta = PB^2-BE^2$
$\displaystyle \Rightarrow ...$

$\displaystyle \Rightarrow h^2 = \frac{b^2}{4(\tan^2\beta-\cot^2\theta)}$ (1)
$\displaystyle AP^2 = h^2\cot^2\alpha = PE^2+AE^2$
$\displaystyle \Rightarrow ...$

$\displaystyle \Rightarrow h^2 = \frac{(2a+b)^2}{4(\cot^2\alpha - \cot^2\theta)}$ (2)
Eliminate $\displaystyle h$ from (1) and (2) and the result follows.

Can you fill in the gaps?

Can you complete it?

Can you show the rest of the method, I just wanna double check mine. Thanks a lot!

5. Hello cloud5

Question 1

$\displaystyle -\tfrac12>\frac{a^2+(a+d)^2-(a+2d)^2}{2a(a+d)}>-1$

$\displaystyle \Rightarrow-\tfrac12>\frac{a^2+(a^2+2ad+d^2)-(a^2+4ad+4d^2)}{2a(a+d)}>-1$

$\displaystyle \Rightarrow-\tfrac12>\frac{a^2-2ad-3d^2}{2a(a+d)}>-1$

$\displaystyle \Rightarrow-\tfrac12>\frac{(a-3d)(a+d)}{2a(a+d)}>-1$

$\displaystyle \Rightarrow-\tfrac12>\frac{a-3d}{2a}>-1$

$\displaystyle \Rightarrow\tfrac12<\frac{3d-a}{2a}<1$

$\displaystyle \Rightarrow a<3d-a<2a$

$\displaystyle \Rightarrow 2a<3d<3a$

$\displaystyle \Rightarrow \tfrac23a<d<a$

Question 2

(ii) $\displaystyle PE^2 = h^2\cot^2\theta =PB^2-BE^2$
$\displaystyle =h^2\cot^2\beta-\tfrac14b^2$
$\displaystyle \Rightarrow h^2(\cot^2\beta-\cot^2\theta) = \tfrac14b^2$

$\displaystyle \Rightarrow h^2=\frac{b^2}{4(\cot^2\beta-\cot^2\theta)}$ (1) (Note the typo in my previous post: not $\displaystyle \tan^2\beta$.)

Also $\displaystyle AP^2 = h^2\cot^2\alpha = PE^2+AE^2$
$\displaystyle =h^2\cot^2\theta+(a+\tfrac12b)^2$
$\displaystyle \Rightarrow h^2(\cot^2\alpha-\cot^2\theta) = (a+\tfrac12b)^2$

$\displaystyle \Rightarrow h^2=\frac{(2a+b)^2}{4(\cot^2\alpha-\cot^2\theta)}$
$\displaystyle =\frac{b^2}{4(\cot^2\beta-\cot^2\theta)}$ from (1)
$\displaystyle \Rightarrow (2a+b)^2(\cot^2\beta-\cot^2\theta)=b^2(\cot^2\alpha-\cot^2\theta)$

$\displaystyle \Rightarrow (2a+b)^2\cot^2\beta-b^2\cot^2\alpha = (4a^2+4ab+b^2)\cot^2\theta - b^2\cot^2\theta$
$\displaystyle =4a(a+b)\cot^2\theta$

(ii) $\displaystyle PE^2 = h^2\cot^2\theta =PB^2-BE^2$
$\displaystyle =h^2\cot^2\beta-\tfrac14b^2$
$\displaystyle \Rightarrow h^2(\cot^2\beta-\cot^2\theta) = \tfrac14b^2$

$\displaystyle \Rightarrow h^2=\frac{b^2}{4(\cot^2\beta-\cot^2\theta)}$ (1) (Note the typo in my previous post: not $\displaystyle \tan^2\beta$.)

Also $\displaystyle AP^2 = h^2\cot^2\alpha = PE^2+AE^2$
$\displaystyle =h^2\cot^2\theta+(a+\tfrac12b)^2$
$\displaystyle \Rightarrow h^2(\cot^2\alpha-\cot^2\theta) = (a+\tfrac12b)^2$

$\displaystyle \Rightarrow h^2=\frac{(2a+b)^2}{4(\cot^2\alpha-\cot^2\theta)}$
$\displaystyle =\frac{b^2}{4(\cot^2\beta-\cot^2\theta)}$ from (1)
$\displaystyle \Rightarrow (2a+b)^2(\cot^2\beta-\cot^2\theta)=b^2(\cot^2\alpha-\cot^2\theta)$

$\displaystyle \Rightarrow (2a+b)^2\cot^2\beta-b^2\cot^2\alpha = (4a^2+4ab+b^2)\cot^2\theta - b^2\cot^2\theta$
$\displaystyle =4a(a+b)\cot^2\theta$
Would you mind showing the sketch of the triangle for Q2? Can you explain why you're using $\displaystyle \cot$ for Q2i ?

7. Hello cloud5
Originally Posted by cloud5
Would you mind showing the sketch of the triangle for Q2? Can you explain why you're using $\displaystyle \cot$ for Q2i ?
Look at the attached diagram.

$\displaystyle AP, BP, EP, CP$ are all horizontal lines, and therefore are perpendicular to $\displaystyle PT$. So from the right-angled triangles, we get:
$\displaystyle AP = h\cot\alpha$

$\displaystyle BP = h\cot\beta$

$\displaystyle EP = h\cot\theta$

$\displaystyle CP = h \cot\beta$
Thus $\displaystyle BP = PC$, and $\displaystyle \triangle BPC$ is therefore isosceles, with $\displaystyle E$ the mid-point of $\displaystyle BC$. Hence:
$\displaystyle \angle PEB = 90^o$

$\displaystyle E$ is therefore the foot of the perpendicular from $\displaystyle P$ to $\displaystyle BC$

Therefore $\displaystyle PE$ is the shortest distance from $\displaystyle P$ to $\displaystyle BC$, and so:

$\displaystyle \theta$ is therefore the largest angle of elevation of $\displaystyle T$ from the line $\displaystyle BC$
The remaining working uses Pythagoras' Theorem on the right-angled triangles $\displaystyle APE$ and $\displaystyle BPE$.