Find the area of the quadrilateral in the figure.
1 answer:
<h3>Answer:</h3>
C. 19.64
<h3>Explanation:</h3>The triangle at upper right is a 3-4-5 right triangle, so has an area that is half the product of the leg lengths:
... upper right area = (1/2)(3 units)(4 units) = 6 units²
The triangle at lower left is an isosceles triangle with base length 5 and side length 6. The altitude to the side of length 5 is a bisector of that side and forms right angles at the point of intersection. Hence we can use the Pythagorean theorem to find the triangle's altitude:
... lower left triangle altitude = √(6² - 2.5²) = √29.75 ≈ 5.45436
Then the area of the lower left triangle is half the product of this altitude and the base length of 5 units:
... lower left area = (1/2)(5.45436 units)(5 units) ≈ 13.6359 units²
The quadrilateral's area is the sum of the areas of these triangles, so is ...
... quadrilateral area = upper right area + lower left area
... = 6 units² + 13.6359 units²
... = 19.6359 units² ≈ 19.64 units²
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Confirmed by my geometry program as shown in the attachment.
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<h3>Answer: 24/25</h3>
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Explanation:
Sine is given to be negative, and so is tangent. This only happens in quadrant Q4
Recall that y = sin(theta), so if sin(theta) < 0, then we're below the x axis.
If tan(theta) < 0, then this means cos(theta) > 0
So we have y < 0 and x > 0 which places the angle somewhere in Q4.
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Draw a right triangle as shown below in the attached image. We have AC = 25 and BC = 7. Use the pythagorean theorem to find that AB = 24
So this is what your steps may look like
a^2+b^2 = c^2
7^2+b^2 = 25^2
b^2+49 = 625
b^2 = 625-49
b^2 = 576
b = sqrt(576)
b = 24
So because AB = 24, we know that the cosine of the angle is adjacent/hypotenuse = 24/25
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As an alternative, you could use the trig identity
sin^2(x) + cos^2(x) = 1
and plug in the given value of sine to solve for cosine. The cosine value result will be positive since we're in Q4.
So,
sin^2(x) + cos^2(x) = 1
(-7/25)^2 + cos^2(x) = 1
(49/625) + cos^2(x) = 1
cos^2(x) = 1 - (49/625)
cos^2(x) = (625/625) - (49/625)
cos^2(x) = (625-49)/625
cos^2(x) = 576/625
cos(x) = sqrt(576/625)
cos(x) = sqrt(576)/sqrt(625)
cos(x) = 24/25
This is effectively a rephrasing of the previous section since the pythagorean trig identity is more or less the pythagorean theorem (just in a trig form)
Answer:
Step-by-step explanation:
We have the compound inequality:
Let's solve each of them individually first:
We have:
Divide both sides by 2:
Add 1 to both sides:
We have:
Subtract from both sides:
Divide both sides by -4:
Hence, our solution set is: