Triangle ABC is given where m∠A=33°, a = 15 in., and the height, h, is 9 in. How many distinct triangles can be made with the given measurements? Explain your answer
To determine the number of distinct triangles that can be made with the given measurements, we need to analyze the given information and consider the constraints imposed by the properties of triangles.
In a triangle, the measures of the angles and the lengths of the sides are related by certain rules. The most important rule is the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's analyze the given information:
1. Angle A (m∠A) is given as 33°.
2. Side a is given as 15 in.
3. The height, h, is given as 9 in.
Since angle A is given, we know the measure of one angle in the triangle. However, this does not uniquely determine the triangle, as the other two angles can vary.
To determine the number of distinct triangles, we need to examine the constraints imposed by the Triangle Inequality Theorem for the given side lengths.
The height (h) of the triangle is perpendicular to side a. This means that height (h) can be considered as the altitude to base a. Using the altitude, we can form a right-angled triangle (in this case, a right-angled triangle with angle A as the right angle) and apply trigonometric ratios such as sine, cosine, and tangent.
Using trigonometry, we can find the lengths of the other sides in the triangle. Since we know the height (h) and angle A, we can use the trigonometric ratio tangent (tan) to find the length of side b (opposite to angle A):
tan(A) = opposite/adjacent
tan(33°) = h/b
Solving for b: b = h/tan(33°)
Using the given h = 9 in. and calculating b, we find b ≈ 9.905 in. (rounded to three decimal places).
Now we have two side lengths, a = 15 in. and b ≈ 9.905 in., and we need to find the range of possible lengths for the remaining side, side c.
We can apply the Triangle Inequality Theorem:
1. The sum of sides a and b must be greater than the third side, c:
a + b > c => 15 + 9.905 > c => 24.905 > c
2. The sum of sides b and c must be greater than the third side, a:
b + c > a => 9.905 + c > 15 => c > 5.095
3. The sum of sides a and c must be greater than the third side, b:
a + c > b => 15 + c > 9.905 => c > -5.095
From these inequalities, we can determine the range of values for side c. Since the length of a side cannot be negative, we can ignore the inequality c > -5.095. Thus, the range for side c is 5.095 < c < 24.905.
To summarize, the length of side a is fixed at 15 in., the height is fixed at 9 in., and the length of side b is approximately 9.905 in. The only side that can vary within the constraints of the Triangle Inequality Theorem is side c, which can take any value between 5.095 and 24.905 (exclusive).
Therefore, the number of distinct triangles that can be formed with the given measurements is infinite since there is an infinite number of real numbers between 5.095 and 24.905. Each value within the range will correspond to a distinct triangle.
In order to determine the number of distinct triangles that can be made with the given measurements, we can use the Triangle Inequality Theorem.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the third side.
Let's analyze the given information:
- Angle A is 33°
- Side a is 15 in.
- The height, h, is 9 in.
Since the height, h, is given, we can determine that side a is the base of the triangle. Now, let's consider the possible ranges for the other two sides of the triangle.
1. Side b:
The Triangle Inequality Theorem tells us that side b must satisfy the following condition:
b < a + 9
b < 15 + 9
b < 24
2. Side c:
Similarly, side c must satisfy:
c < a + 9
c < 15 + 9
c < 24
To find the upper limits for side b and side c, we need to consider the angles that are opposite to these sides.
Angle B:
The sum of the angles in a triangle is always 180°.
So, m∠B = 180° - m∠A - m∠C
m∠B = 180° - 33° - m∠C
m∠B = 147° - m∠C
Since m∠B and m∠C are both acute angles (less than 90°), we know that m∠B must be greater than 33°.
Angle C:
m∠C = 180° - m∠A - m∠B
m∠C = 180° - 33° - m∠B
m∠C = 147° - m∠B
Similarly, m∠C must be greater than 33°.
Considering all the conditions, we have:
33° < m∠B < 147°
33° < m∠C < 147°
Now, let's evaluate the possible ranges for sides b and c:
1. Side b:
b < 24 (Upper limit based on the Triangle Inequality Theorem)
33° < m∠B < 147° (Angle B must be greater than 33°)
From trigonometric ratios, we know that the shorter side is always opposite to the smaller angle.
Therefore, b > 15 * sin(33°)
b > 8.45 (approx.)
2. Side c:
c < 24 (Upper limit based on the Triangle Inequality Theorem)
33° < m∠C < 147° (Angle C must be greater than 33°)
Similarly, c > 15 * sin(33°)
c > 8.45 (approx.)
Thus, side b and side c must be greater than 8.45 inches in order to form a triangle with the given measurements.
In summary, there is a range of values for side b and side c that will satisfy the given conditions, and therefore, there are infinitely many distinct triangles that can be made with the given measurements.
two
draw a right triangle with angle A=33° and altitude 9.
That makes the hypotenuse AC = 16.5, and the base AB = 10.7
We want side a (BC) to be 15.
So, draw a circle with center at C and radius 15. It cuts the extended side AB in two places, either of which could be vertex B.