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3 years ago

In triangle △ABC, ∠ABC=90°, BH is an altitude. Find the missing lengths. AH=HC+2 and BC=2, find HC.

Mathematics

2 answers:

Sergio039 [100]3 years ago

4 0

Answer:

HC = 1

Step-by-step explanation:

AC = HC +AH = HC +(HC+2) = 2·HC +2

The altitude divides the triangle into similar triangles, so the ratio of hypotenuse to short side is the same for all. That is ...

BC/HC = AC/BC

2/HC = (2HC +2)/2

4 = 2(HC)(HC +1) . . . . . cross multiply

0 = HC² +HC -2 . . . . . . divide by 2, subtract 2

0 = (HC -1)(HC +2) . . . . factor. Solutions are those values of HC that make the factors be zero.

The useful solution is ...

HC = 1

mylen [45]3 years ago

4 0

Answer:

The measure of HC is 1 unit.

Step-by-step explanation:

Given,

In triangle ABC,

∠ABC = 90°, BC = 2,

Also, H∈ AC such that ∠BHC = 90°,

And, AH = HC + 2

We have to find : HC

∵ ∠ABC = ∠BHC ( right angles )

∠ACB = ∠HCB

By the AA similarity postulate,

$\triangle ABC\sim \triangle BHC$

∵ The corresponding sides of similar triangles are in same proportion,

$\implies \frac{BC}{HC}=\frac{AC}{BC}$

$\frac{2}{HC}=\frac{AH+HC}{2}$

$4=HC(HC+2+HC)$

$4=HC(2HC+2)$

$4=2HC(HC+1)$

$2=HC(HC+1)$

$\implies HC^2+HC-2=0$

$HC^2+2HC-HC-2=0$

$HC(HC+2)-1(HC+2)=0$

$(HC+2)(HC-1)=0$

By zero product property,

HC = -2 ( not possible ) or HC = 1

Hence, the measure of HC is 1 unit.

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Scott likes to run long distances. He can run 20 \text{ km}20 km20, start text, space, k, m, end text in 858585 minutes. He want

aleksandrvk [35]

Answer:

Scott will take 221 minutes to run 52 km

Step-by-step explanation:

Speed

The speed of an object can be calculated with the formula:

$\displaystyle v=\frac{d}{t}$

Where d is the distance traveled and t is the time taken.

Scott can run d=20 km in 85 minutes. Thus, his speed is:

$\displaystyle v=\frac{20}{85}\ km/min$

Now he wants to know how many minutes it will take him to run d=52 km. Solving the formula for t:

$\displaystyle t=\frac{d}{v}$

Since the speed has been already determined:

$\displaystyle t=\frac{52}{\frac{20}{85}}$

Multiplying by the reciprocal of the denominator:

$\displaystyle t=52*\frac{85}{20}$

t = 221 min

Scott will take 221 minutes to run 52 km

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3 years ago

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Anni [7]

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The tangent to the circle x^2+y^2=18 is parallel to the tangent x+y=6

liubo4ka [24]

Answer:

y = -x - 6

Step-by-step explanation:

If we are looking for the equation of the line tangent to the circle, we have to find the first derivative of the circle function. If that tangent line is to be parallel to the given tangent line, we need to find the slope of the the given tangent line and make sure the slope of the line we are looking for is the same. First let's find the slope of the given tangent. If the given tangent is x+y=6, then to find the slope, solve for y:

y = -x + 6. So the slope of that line, and also the slope of the tangent we are solving for, is -1. Hold that thought while we find the derivative of the function. Using implicit differentiation, we find the derivative to be:

$2x+2y\frac{dy}{dx}=0$

Solving for the slope (dy/dx) gives us:

$2y\frac{dy}{dx}=-2x$ and

$\frac{dy}{dx}=\frac{-2x}{2y}$ so

$\frac{dy}{dx}=-\frac{x}{y}$

Subbing in the value of the slope we found:

$-1=-\frac{x}{y}$ so

-y = -x or, equivalently,

y = x. Now that we know that y and x are the same value, we can go back to the original circle to find out what they both are by substitution. If y = x, we make the substution:

If $x^2+y^2=18$ , then

$y^2+y^2=18$ and

$2y^2=18$ and

$y^2=9$ so

y = ±3

We will choose the negative root (you'll see why in a second) and sub that in for both y and x since y and x are th same number. Going back to the fact that the slope is -1:

y - (-3) = -1(x - (-3)) simplifies a bit to:

y + 3 = -x - 3 which gives us, in slope-intercept form:

y = -x - 6

That is the equation of the tangent to the circle that is parallel to the given tangent.

If we would have chosen the principle (or positive) root of 3, our equation would have looked like this:

y - 3 = -1(x - 3) and

y = -x + 3 + 3 so

y = -x + 6. Notice that that is the EXACT SAME EQUATION as the given tangent. That's why we have to pick the negative 3 as our root.

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