Find bc, if the area of the triangle abc is 36 cm 2 and the height ad is 3 cm.

What if I solve this by saying that Triangle ABC is congruent to itself (through SAS) in this way - 1. AC congruent to AB (Symmetric Property) 2. Angle A congruent to Angle A (Reflexive) 3. Triangle ABC congruent to Triangle ABC (SAS) 4. So Angle B congruent to Angle C (CPCTC) Is this an acceptable way of proving it? Yes, that is a very good strategy indeed. That is called Pappus' proof, because Euclid didn't think of it when he wrote The Elements. (Euclid didn't use Sal's proof either, because this is Proposition 5 and SSS doesn't get proved until Proposition 8.) It's much simpler once you get over the initial hump of how weird it is to have a congruence proof with what looks like only one triangle. There is one change you should make to the proof, though. You should make your point in step 3 more clear by saying that Triangle ABC is congruent to triangle ACB -- you see how I lined up the letters to make it clear what the corresponding vertices are? Because, to be honest, you could show that any triangle is congruent to itself whether it is isosceles or not. Equality (=) is used for measurements: length (inches, cm, etc.), angle measures (degrees), area (cm^2) Congruency (= with a ~ over it) is used for objects: line segments, angles, polygons, circles. These are things that have the all the same properties (equal measurements.) Orientation is not important, so you can rotate or flip a polygon and it can still be congruent. But, these objects are not numbers. Much like th...

Step1: Finding the area of Δ A B C ∆ A B C is right–angled at A. A B = 5 c m , B C = 13 c m and A C = 12 c m now, area of triangle ∆ A B C = 1 2 × a l t i t u d e × b a s e = 1 2 × A B × A C = 1 2 × 5 c m × 12 c m = 30 c m ² Hence, area of ∆ A B C = 30 c m ² Step2:Finding length of A D Area of triangle A ∆ A B C = 1 2 × B C × A D ⇒ 30 = 1 2 × 13 × A D ⇒ 30 × 2 13 = A D ⇒ A D = 60 13 ⇒ A D = 4 . 61 c m Therefore the area of ∆ A B C = 30 c m ² and length of A D = 4 . 61 c m

First draw the height - due to the symmetry of equilateral triangles, it will start at the midpoint of whichever side you choose to start from, and end at the opposite vertex (point/corner). In effect you will split the equilateral triangle into two congruent right triangles. Now consider one of these two right triangles by itself. If the equilateral triangle has sides of length x, then the hypotenuse of our right triangle will also be x. We also know that the side opposite the 30 degree angle has length x/2, since we split that side of the triangle in half to construct this right triangle. Since you know two of the sides of a right triangle, you can use the Pythagorean theorem to find the length of the 3rd. (x/2)^2 + m^2 = x^2 x^2/4 + m^2 = x^2 m^2 = (3*x^2)/4 m = (x*sqrt(3))/2 Where m is the height of the right triangle, which is equal to the height of the equilateral triangle. Incidentally, this derivation also proves the shortcut for the ratio of the sides in a 30-60-90 triangle, since the effect of cutting an equilateral triangle in half is to create 2 30-60-90 triangles. We don't know what the height is without working it out. Pythagoras' theorem says that in a right triangle, the area of a square drawn on the longest side (the hypotenuse) is equal in area to the areas of the two squares drawn on the other two sides (the legs), added together. The square on the side with length s/2 is 1/4 of the area of the square on the side with length s, so a square on the height ...

Here is an answer from Hadriel Ramos (check the comments!) :Actually it is one of the most fundamental theorems in basic geometry. Just imagine a tall post (10 meters High) and you are standing 20 meters away. And you wanted to tie a rope to the high point of the post to the ground from which you are standing. You use the Pythagorean Theorem. Well, we'll start by defining the sides. one unknown side will be x, and one will be 34 - x. Now, the Pythagorean theorem says that x^2 + (34 - x)^2 = 26^2. Multiplying out, we get: x^2 + 34^2 + x^2 - 68x = 676 Simplifying: 2x^2 - 68x + 1156 = 676 2x^2 - 68x + 480 = 0 x^2 - 34x + 240 = 0 Now, we have a quadratic equation, which we can factor: Now we can solve for x by dividing by x - 10: (x - 24)(x - 10) = 0 x - 24 = 0 And by dividing by x - 24: x - 10 = 0 So, we have that x = 10, and x = 24. This means that the answer to your problem is [24, 10]. - [Tutor] Pause this video and see if you can find the area of this triangle, and I'll give you two hints. Recognize, this is an isosceles triangle, and another hint is that the Pythagorean Theorem might be useful. Alright, now let's work through this together. So, we might all remember that the area of a triangle is equal to one half times our base times our height. They give us our base. Our base right over here is, our base is 10. But what is our height? Our height would be, let me do this in another color, our height would be the length of this line right over here. So, if we can figure...