In ΔABC, m∠ACB = 90°,
CD⊥ AB and m∠ACD = 45°
Find :AC, if CD = 6 sqrt2 in.
If you make your sketch, it is easy to see that ∠CAD is also 45°
giving you an isosceles right-angled triangle, thus
AC^2 = (6√2)^2 + (6√2)^2
= 72+72 = 144
AC = 12
the one above is correct for those of you wondering
Well, well, well, let's solve this geometry puzzle, shall we?
We have a right triangle ΔABC with ∠ACB = 90°. And we know that CD is perpendicular to AB and ∠ACD = 45°.
First, let's take a look at triangle ΔACD. We have a right angle at D and an angle of 45° at ACD, which means that the remaining angle ∠DAC measures 180° - 90° - 45° = 45°.
Since we have a 45°-45°-90° triangle, we know that the sides are in a ratio of 1:1:√2. So, the ratio between AC and CD is 1:√2.
Given that CD measures 6√2 inches, we can multiply it by √2 to find out the length of AC.
AC = CD * √2 = 6√2 * √2 = 6 * 2 = 12 inches.
Thus, AC measures 12 inches. Yippee!
To find AC, we can use trigonometry and the knowledge that CD is the altitude of right triangle ABC.
Let's start by labeling the given information on the triangle:
- ΔABC with a right angle at C
- CD is the altitude from vertex C to side AB, which forms a right angle with AB.
- Angle ACD measures 45 degrees.
- CD = 6√2 inches
To find AC, we can use the trigonometric ratio of sine (sin). In a right triangle, the sine of an acute angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
In this case, AC is the hypotenuse, and AC is opposite angle C. So we can write the equation:
sin(45°) = CD / AC
Now substitute the values:
sin(45°) = (6√2) / AC
Using the trigonometric identity sin(45°) = √2 / 2:
√2 / 2 = (6√2) / AC
To solve for AC, cross-multiply:
2 * (6√2) = AC * (√2 / 2)
12√2 = AC * (√2 / 2)
Multiplying both sides by 2 / √2:
12√2 * (2 / √2) = AC
Simplifying:
12 * 2 = AC
AC = 24
So, AC is equal to 24 inches.
To find AC, we need to use trigonometry and the Pythagorean theorem.
First, let's construct a right triangle with AC as the hypotenuse and CD as one of the legs. Angle ACD is a right angle because CD is perpendicular to AB.
Since m∠ACD = 45°, we know that this is a special right triangle, specifically a 45-45-90 triangle. In a 45-45-90 triangle, the two legs are congruent, and the length of the hypotenuse is sqrt(2) times the length of each leg.
In our case, CD is one of the legs, and we know that CD = 6 sqrt(2) in. Therefore, the length of the other leg (AC) is also 6 sqrt(2) in.
So, AC = 6 sqrt(2) in.