In triangle A B C, ∠A=40^∘, a=15, and b=20 (a) Show that there are two triangles, A B C and A^'

In triangle A B C, ∠A=40^∘, a=15, and b=20 (a) Show that...

Key Concepts

Ambiguous Case (SSA Condition)

The ambiguous case arises in triangle problems when two sides and a non-included angle (SSA) are given. This situation can lead to two distinct triangles, one triangle, or no triangle at all, depending on the relationships among the given measurements. Understanding this case involves determining when and why two different configurations are possible due to the fact that the sine function has the same value for an angle and its supplement, thereby leading to an ambiguity in the solution for the missing angles.

Law of Sines

The Law of Sines establishes the proportional relationship between the lengths of sides of a triangle and the sines of their opposite angles. It is a fundamental tool in trigonometry that is especially useful in solving triangles when certain combinations of sides and angles are known, such as in the SSA case. In the context of the problem, it helps determine the possible measures for the unknown angles and provides insight into the relationship between the side lengths and angles in the ambiguous case.

Area Formula Using Sine

The area of a triangle can be calculated as one-half the product of two sides multiplied by the sine of the included angle. This formula is particularly useful when the measure of the angle between two known sides is available. In problems like this one, it also plays a crucial role in comparing the areas of two triangles by relating the area directly to the sine of the corresponding angles, thus establishing the proportionality between areas and the sine values of specific angles.

Transcript

00:01 For question number 31st, first we're going to start by drawing the triangle and actually showing that there are two triangles that are possible for this question.

00:08 First, we're going to use the fact that measure of angle a is equal to 40 degrees.

00:12 We know that angle a is between sides b and c.

00:15 We don't know what is the measure of side c.

00:17 So we're just going to draw a random line representing side c.

00:21 But we know that side b measures 20 degrees, 20 units.

00:25 So we're going to draw a line with 20 units as a measure that makes a 40 degree angle with side c and.

00:31 And that's going to be our side b, and here is going to be angle a.

00:35 So now that we have drawn everything, the only thing that is missing right now is where is side a itself.

00:42 You know that side a should start from here.

00:44 So we're going to draw a circle using this point as its origin and using 15 as its radius because that's a measure of side a.

00:54 And we're going to draw a circle.

00:55 When we draw a circle, we can see that the circle would intersect with side c in two points.

01:00 So basically there are two side a's that are possible, and that's proved the fact that there are actually two triangles that satisfy the given conditions.

01:07 I'm going to draw each of them separately so we could see better.

01:10 One of them that i'm going to name as a, b, b prime, c prime is the small triangle on the left that we have a as here, and we're going to have c here and here is b.

01:22 But again, there are a prime, b prime, and c prime.

01:24 So we know that the measure of b prime, like side b is actually equal to 20 units.

01:30 Measure of side a is equal to 15 units.

01:33 And here we have 40 degree as the angle a.

01:36 The other triangle is going to be the bigger triangle.

01:40 Basically for bigger triangle, again, we have side a, the angles a, b, and c.

01:45 We have side a, which measures 15 units inside.

01:49 Here is, sorry, i put them in the opposite part.

01:54 So here is going to be angle b and here is angle c, and we know that side b measures 20 units, and angle a is actually equal to 40 degree.

02:04 So now that we have drawn each of this, let's solve for the part b.

02:10 For part b, it says that we should show that the proportionality of the areas is equal to the proportionality of the sign of the angle c.

02:20 So let's see.

02:21 First of all, i'm going to try, because we need to find out the area, so i'm going to draw the height in each of the triangles.

02:29 For this one, the height is going to be here.

02:32 So i'm going to name it as h prime, and this one, the height is here, and i'm going to name it as h.

02:38 Let's see which each of them is equal to what.

02:41 So h prime, if we consider the big triangle on the left, so basically we have the sign of a prime is actually equal to the h prime over 20.

02:53 So that means that h prime is actually equal to sign of a prime and a prime is 40 degrees...

Please give Ace some feedback