How is the formula for a triangle derived from a parallelogram? Please give a good explanation.

Mathematics

1 answer:

Katen [24]3 years ago

5 0

Answer:

At first, we divide the parallelogram into two triangles by joining any two opposite vertices. These two triangles are exactly the same (congruent) and thus have equal areas. The area of the parallelogram is the summation of the individual areas of the two triangles. We drop a perpendicular from a vertex to its opposite side to get an expression for the height of the triangles. The area of the individual triangle is 12×base×height12×base×height .The area of the parallelogram being twice the area of the triangle, thus becomes after evaluation base×heightbase×height .

Complete step by step answer:

The parallelogram can be divided into two triangles by constructing a diagonal by joining any two opposite vertices.

In the above figure, ΔABDΔABD and ΔBCDΔBCDare the two such triangles. These two triangles have:

AB=CDAB=CD (as opposite sides of a parallelogram are equal)

AD=BCAD=BC (opposite sides of a parallelogram are equal)

BDBD is common

Thus, the two triangles are congruent to each other by SSS axiom of congruence. Since, the areas of two congruent triangles are equal,

⇒area(ΔABD)=area(ΔBCD)⇒area(ΔABD)=area(ΔBCD)

Now, we need to find the area of ΔABDΔABD . We draw a perpendicular from DD to the side ABAB and name it as DEDE . Thus, ΔABDΔABD is now a triangle with base ABAB and height DEDE .

Then, the area of the ΔABD

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daser333 [38]

Step-by-step explanation:

Notice the two horinzontal lines. They are parallel lines as they will never cross each other.

Notice the slant line. That is called a transversal as it crosses through both the parallel lines.

The transversal split the parallel lines into multiple angles.

Vertical Angles
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Linear Pair Angles
Same Side Consecutive Angles

Look at Angle X and the angle on the upper right. They are vertical angles because they are formed by intersecting lines so they are equal to each other

Look at Angle X and the angle that is on the right of Angle 70, they are alternate interior angles as they are formed by the transversal crossing through the two parallel lines. They are equal to each other. They form a Z if you connect a them.

The upper right angle and bottom left angle are alternate exterior angles. They are formed by the transversal. They are equal to each other. They form a semi circle of you connect them.

The bottom left angle and Angle X are Corresponding Angles. They form a C shape if you connect them. They are formed by the transversal and are congruent to each other.

Angle X and 70 are same side consecutive angles. They are right next to each other on the same side of each other. They are supplementary. which means they form 180 degrees when added.

so using this info,

$x + 70 = 180$