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## What is the value of x according to the given lengths of angle bisectors?

sankarankalyanam
Featured 1 week ago

(a) Perimeter of the triangle ABC

$P = \left(a + b + c\right) = 80.68 + 72 + 76.34 \textcolor{b l u e}{= 229.02}$

#### Explanation:

(a)
${b}_{1} / {b}_{2} = \frac{{a}_{1} + {a}_{2}}{{c}_{1} + {c}_{2}}$ Eqn (1)

${a}_{1} / {a}_{2} = \frac{{c}_{1} + {c}_{2}}{{b}_{1} + {b}_{2}}$ Eqn (2)

${c}_{1} / {c}_{2} = \frac{{b}_{1} + {b}_{2}}{{a}_{1} + {a}_{2}}$ Eqn (3)

Multiplying Eqn (1) by (3),

$\left({b}_{1} / {b}_{2}\right) \cdot \left({c}_{1} / {c}_{2}\right) = \frac{\frac{\cancel{{a}_{1} + {a}_{2}}}{{c}_{1} + {c}_{2}}}{\frac{{b}_{1} + {b}_{2}}{\cancel{{a}_{1} + {a}_{2}}}}$

$\frac{{c}_{2} \cdot {b}_{1}}{{c}_{1} \cdot {b}_{2}} = \frac{{b}_{1} + {b}_{2}}{{c}_{1} + {c}_{2}}$

$\left({c}_{2} \cdot {b}_{1}\right) \cdot \left({c}_{1} + {c}_{2}\right) = \left({b}_{1} + {b}_{2}\right) \cdot \left({c}_{1} \cdot {b}_{2}\right)$

${c}_{2}^{2} {b}_{1} + {c}_{2} {c}_{1} B - 1 = {b}_{1} {b}_{2} {c}_{1} + {b}_{2}^{2} {c}_{1}$

Substituting values of ${b}_{1} , {b}_{2} , {c}_{1}$ in the above equation,

$32 {c}_{2}^{2} + 32 \cdot 36 \cdot {c}_{2} = 32 \cdot 40 \cdot 36 + {40}^{2} \cdot 36$

${\cancel{32}}^{1} {c}_{2}^{2} + {\cancel{1152}}^{36} {c}_{2} = {\cancel{46080}}^{1280} + {\cancel{57600}}^{1800}$

${c}_{2}^{2} + 36 {c}_{2} - 3080 = 0$

${c}_{2} = \frac{- 36 \pm \sqrt{{36}^{2} + 4 \cdot 3080}}{2} = \frac{- 36 + 116. .69}{2} = 40.34$ rounded to two decimals and leaving the negative value.

Considering Equation (3),

${c}_{1} / {c}_{2} = \frac{{b}_{1} + {b}_{2}}{{a}_{1} + {a}_{2}}$

$\frac{36}{40.34} = \frac{40 + 32}{a}$ where $a = \left({a}_{1} + {a}_{2}\right)$

$a = \frac{72 \cdot 40.34}{36} = 80.68$

Considering Eqn (2),

$\frac{a - {a}_{2}}{a} _ 2 = \frac{{c}_{1} + {c}_{2}}{{b}_{1} + {b}_{2}} = \frac{76.34}{72}$

#80.68 - a_2 = (76.34/72) * a_2

$148.34 {a}_{2} = 80.68 \cdot 72$

${a}_{2} = \frac{80.68 \cdot 72}{148.34} = 39.16$

${a}_{1} = 80.68 - {a}_{2} = 80.68 - 39.16 = 41.52$

Perimeter of the triangle ABC

$P = \left(a + b + c\right) = 80.68 + 72 + 76.34 \textcolor{b l u e}{= 229.02}$

Similarly, we can find P for Question (b)

## A right triangle has a hypotenuse of 30 inches. Both of its legs are the same length. How long is a leg of this right triangle?

sankarankalyanam
Featured 1 week ago

Length of a leg $\vec{A C} = \vec{B C} = \textcolor{g r e e n}{21.21}$

#### Explanation:

In triangle ABC in the above figure' hypotenuse vec(AB) = 30"

$\hat{B} , \hat{A}$ are equal and $\hat{C} = {90}^{0}$

$\therefore \hat{B} = \hat{A} = \frac{180 - 90}{2} = {45}^{0}$

Using trigonometric functions,

#vec(AC) = vec(AB) * sin B = 30 * sin(45) = 30 * (1/sqrt2) = 21.21"#

#vec(BC) = vec(AB) = 21.21"#

## Find the equation of the circle which passes through the origin and #(4, −8)#? #\ \ \ \ # Why do we use midpoint of two points to calculate the equation of circle? The points might make a chord rather than a diameter! Give me some insight?

Jim H
Featured 6 days ago

#### Explanation:

I did not use the midpoint at all.

Using the equation of a circle:

${x}^{2} - 2 h x + {h}^{2} + {y}^{2} - 2 k y + {k}^{2} = {r}^{2}$

Because it passes through $\left(0 , 0\right)$, we find the ${r}^{2} = {h}^{2} + {k}^{2}$.

So, ${x}^{2} - 2 h x + {y}^{2} - 2 k y = 0$

Using the point $\left(4 , - 8\right)$, we get

$16 - 8 h + 64 + 16 k = 0$

So, $h = 2 k + 10$

Every circle that contains the points $\left(0 , 0\right)$ and $\left(4 , - 8\right)$ has equation:

${\left(x - h\right)}^{2} + {\left(y - k\right)}^{2} = {h}^{2} + {k}^{2}$ where $h = 2 k + 10$

## Find the ratio in which the staight line 5x+4y=4 divides the join of (4,5) and (7,-1)?

Shiva Prakash M V
Featured 4 days ago

Hence, the ratio in which the point divides the line is
$4 : - 3 = - 4 : 3$

#### Explanation:

We need to find the point of intersection of the lines
$5 x + 4 y = 4$ and
$\frac{y - 5}{x - 4} = \frac{- 1 - y}{7 - x}$

Simplifying the second line

$\left(y - 5\right) \left(7 - x\right) = \left(- 1 - y\right) \left(x - 4\right)$

$\left(y - 5\right) \left(x - 7\right) - \left(y + 1\right) \left(x - 4\right) = 0$

Expanding
$x y - 7 y - 5 x + 35 - \left(x y - 4 y + x - 4\right) = 0$

$x y - 7 y - 5 x + 35 - x y + 4 y - x + 4 = 0$

Rearanging and simplifying
$- 6 x - 3 y + 39 = 0$
Dividing by 13
$- 2 x - y + 13 = 0$

or

$2 x + y = 13$

The equations are:

$5 x + 4 y = 4$-------(1)
$2 x + y = 13$-------(2)

$\det A = 5 \times 1 - 2 \times 4 = 5 - 8 = - 3$
$\det x = 4 \times 1 - 13 \times 4 = 4 - 52 = - 48$
$\det y = 5 \times 13 - 2 \times 4 = 65 - 8 = 57$

$x = \det \frac{x}{\det} A = - \frac{48}{-} 3 = 16$
$y = \det \frac{y}{\det} A = \frac{57}{-} 3 = - 19$

$\left(x , y\right) \equiv \left(16 , - 19\right)$

Check

$5 x + 4 y = 5 \times 16 + 4 \times \left(- 19\right) = 80 - 76 = 4$

$2 x + y = 2 \times 16 + \left(- 19\right) = 32 - 19 = 23$

Verified.

Hence, the intersection point is $P \equiv \left(16 , - 19\right)$
One end of the line is
Other end of the line is $B \equiv \left(7 , - 1\right)$
Arranging in the form of $A P B$
$\left({x}_{A} , {y}_{A}\right) \equiv \left(4 , 5\right)$
$\left({x}_{P} , {y}_{P}\right) \equiv \left(16 , - 19\right)$
$\left({x}_{B} , {y}_{B}\right) \equiv \left(7 , - 1\right)$
P divides AB in the ratio
$\frac{{x}_{P} - {x}_{A}}{{x}_{B} - {x}_{P}} = \frac{16 - 4}{7 - 16} = \frac{12}{-} 9 = \frac{4}{-} 3$along x axis
$\frac{{y}_{P} - {y}_{A}}{{y}_{B} - {y}_{P}} = \frac{- 19 - 5}{- 1 - \left(- 19\right)} = \frac{- 24}{18} = \frac{- 4}{3}$along y axis
Check:
Both are same
Justifying the coordinate for intersection point

Hence, the ratio in which the point divides the line is
$4 : - 3 = - 4 : 3$

## A circle has a radius of 3 feet and a central angle DEF that measures 60°. What is the length of the intercepted arc DF?

Jim G.
Featured 2 days ago

$\text{arc DF "=pi~~3.14" feet to 2 dec. places}$

#### Explanation:

$\text{the length of the arc is calculated using}$

#• " length of arc "="circumference "xx"fraction of circle"#

$\textcolor{w h i t e}{\times \times \times \times \times \times} = 2 \pi r \times \frac{60}{360}$

$\textcolor{w h i t e}{\times \times \times \times \times \times} = 6 \pi \times {\cancel{60}}^{1} / {\cancel{360}}^{6}$

$\textcolor{w h i t e}{\times \times \times \times \times \times} = \frac{\cancel{6} \pi}{\cancel{6}} = \pi \approx 3.14$

## A triangle with three degree angle ratio is 4:3:2, this is what the triangle triangle?

Jim G.
Featured 2 days ago

${80}^{\circ} , {60}^{\circ} \text{ and } {40}^{\circ}$

#### Explanation:

$\text{sum the parts of the ratio }$

$\Rightarrow 4 + 3 + 2 = 9 \text{ parts}$

#• " the sum of the 3 angles in a triangle "=180^@#

$\Rightarrow {180}^{\circ} / 9 = {20}^{\circ} \leftarrow \textcolor{b l u e}{\text{1 part}}$

$\Rightarrow 4 \text{ parts } = 4 \times {20}^{\circ} = {80}^{\circ}$

$\Rightarrow 3 \text{ parts } = 3 \times {20}^{\circ} = {60}^{\circ}$

$\Rightarrow 2 \text{ parts } = 2 \times {20}^{\circ} = {40}^{\circ}$

$\text{the 3 angles in the triangle are}$

${80}^{\circ} , {60}^{\circ} \text{ and } {40}^{\circ}$

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